Writing the equation of a line from its slope and y-intercept, and from a graph. Includes reading a y-intercept off a graph and computing the slope from two points on it, handling axes drawn to different scales, and modelling a falling quantity with a negative slope.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 5 — Writing Linear Equations
Slope-Intercept Form
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.1 Slope-Intercept Form §5.1, pp. 269-276 — the lesson these objectives are drawn from
Warm-up
Lesson 4.7 read the slope and y-intercept out of an equation. Chapter 5 runs the same machinery backwards.
Discussion prompt
Given the equation y equals 3x minus 4, you can name the slope and intercept in a second. Now suppose you were told the slope is 3 and the intercept is negative 4 — could you recover the equation, and how?
Hint: Look at what the form is made of.
Answer:
\[ y = mx + b \;\xrightarrow{\; m = 3, \; b = -4 \;}\; y = 3x - 4 \]
The form has exactly two blanks in it, and being told the two numbers fills both. Every question in this chapter is a version of that: find the two numbers by some route, then write them into the form.
Concept
The slope-intercept form of the equation of a line with slope m and y-intercept b is y equals mx plus b. If you know those two numbers, you can write the equation without graphing anything.
slope-intercept form — The form y equals mx plus b of a linear equation, where m is the slope of the line and b is its y-intercept.
The whole of Chapter 5 is about finding m and b from whatever information a question happens to give you.
Figure (svg): Two given numbers substituted into the slope-intercept form to produce an equation
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.1 Slope-Intercept Form §5.1, pp. 269-269
Section
Section 1
Concept
When the slope and the y-intercept are both given, writing the equation is a single substitution into the form. Nothing has to be solved or graphed.
\[ m = 3, \; b = -4 \;\Longrightarrow\; y = 3x - 4 \]
Figure (svg): Two given numbers substituted into the slope-intercept form to produce an equation
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.1 Slope-Intercept Form §5.1, pp. 269-269 — Example 1, Equation of a Line
Picture it
The form is a template with two blanks.
Figure (svg): Two given numbers substituted into the slope-intercept form to produce an equation
The two numbers go into different places and mean different things, so knowing which is which matters more than the substitution itself.
Worked example
This is Example 1 from the textbook.
\[ \text{Write the equation of the line whose slope is } 3 \text{ and whose } y\text{-intercept is } -4. \]
Write the slope-intercept form
Why: The template with two blanks.
\[ y = m x + b \]
Substitute the slope for m
Why: Three replaces m.
\[ y = 3 x + b \]
Substitute the intercept for b
Why: Write it in brackets to keep its sign.
\[ y = 3 x + (-4) \]
Simplify
Why: Adding negative four is subtracting four.
\[ y = 3 x - 4 \]
Figure (svg): A negative intercept substituted into the form and then simplified
\[ y = 3x - 4 \]
Verify: read the two numbers back out of the answer
Why: The coefficient of x is three and the constant is negative four, which are the numbers given. Reading the finished equation back the way Lesson 4.7 taught is a genuine check, since it uses the reverse process rather than repeating the same one.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.1 Slope-Intercept Form §5.1, pp. 269-269
Matching
The slope multiplies x; the intercept stands alone.
Match the pairs
Why: The first two use the same pair of numbers in opposite roles and give two entirely different lines. That pair is worth studying rather than just matching: one rises gently from below the origin and the other falls steeply from above it.
Worked example
Guided Practice 1 and 2. One has a whole slope and one a fractional one.
\[ \text{Slope } 2 \text{ with intercept } 7, \text{ and slope } \tfrac{2}{5} \text{ with intercept } 6. \]
Take the first pair
Why: Two for m and seven for b.
\[ y = 2 x + 7 \]
Take the second pair
Why: Two fifths for m and six for b.
\[ y = (\frac{2}{5}) x + 6 \]
Check the fraction is on the right term
Why: The slope multiplies x; the intercept stands alone.
Note what did not change
Why: A fractional slope needs no different treatment.
Figure (svg): The solution to Worked example two from guided practice shown as a ladder of expressions, one row per algebraic move
\[ y = 2x + 7 \qquad y = \tfrac{2}{5}x + 6 \]
Verify: test each equation at x equal to zero
Why: Both give the stated intercept, seven and six respectively, since the x-term vanishes. That check confirms the two numbers went into the right slots, which is the only thing that can really go wrong here.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.1 Slope-Intercept Form §5.1, pp. 269-269
Trap
The slope is 3 and the y-intercept is -4.
Write y = -4x + 3
Why: Both numbers appear and the form looks right, so the answer feels finished.
That line has slope negative four and intercept three — a completely different line, falling steeply from above the origin rather than rising from below it.
\[ y = 3x - 4 \quad m = 3, \; b = -4 \]
Attach each number to its letter before writing anything
Why: The slope multiplies x and the intercept stands alone, which is what distinguishes the two positions.
Substituting x equal to zero into the finished equation gives the intercept back, which catches a swap in one line.
Faded example
Substitute, then simplify.
Fill in the blanks
m = 3, \; b = -4: \quad y = 3x + (-4) = 3x - 4
Why: The slope goes in front of x and the intercept goes on its own, brackets included so the negative sign survives. Simplifying then turns plus negative four into minus four.
Elimination
Check both numbers, not just one.
Eliminate the wrong options
Which is the right equation?
Survives elimination: A
Why: The slope multiplies x and keeps its negative sign, and the intercept stands alone as positive five. Three of these four contain both digits, which is why checking each number against its own role is necessary rather than glancing at whether the numbers look familiar.
Socratic
Not one, and not three.
Discussion prompt
Explain why a line needs exactly two numbers to determine it, connecting this to the fact that two points determine a line. Then say what a direct variation model from Lesson 4.6 needs and why it needs less.
Hint: Count the blanks in the form.
Answer:
The form has two blanks, so two pieces of information fill it. That matches the geometric fact that two points determine a line: two points give two facts, and so do a slope and an intercept — they are two different ways of supplying the same amount of information.
A direct variation model is y equals kx, with only one blank, because the intercept is already known to be zero. One piece of information fills it, which is why a single pair of values was enough in Lesson 4.6. Assuming the line passes through the origin is itself the second piece of information, supplied in advance.
Section
Section 2
Concept
When a graph clearly shows where the line crosses the vertical axis, the y-intercept can be read off directly. Only the slope then has to be computed.
\[ \text{crossing at } (0, 2) \;\Longrightarrow\; b = 2 \]
The y-intercept is the y-coordinate of the point where the line crosses the y-axis, so a crossing at (0, 2) gives b equal to 2.
Figure (svg): A graphed line with its y-intercept and a rise-run triangle marked
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.1 Slope-Intercept Form §5.1, pp. 270-270 — the Equations from Graphs paragraph and its Study Tip
Picture it
Half the work is already done.
Figure (svg): A graphed line with its y-intercept and a rise-run triangle marked
Using the intercept as one of the two points for the slope means the run is measured from zero, which makes the subtraction in the denominator trivial.
Worked example
This is Example 2 from the textbook.
\[ \text{A line crosses the } y\text{-axis at } (0, 2) \text{ and passes through } (5, 6). \text{ Write its equation.} \]
Write the form
Why: The template to be filled.
\[ y = m x + b \]
Compute the slope from the two points
Why: Six minus two over five minus zero.
\[ \frac{4}{5} \]
Read the intercept off the graph
Why: The line crosses the vertical axis at (0, 2).
\[ b = 2 \]
Substitute both numbers
Why: Four fifths for m and two for b.
\[ y = (\frac{4}{5}) x + 2 \]
Figure (svg): A graphed line with its y-intercept and a rise-run triangle marked
\[ y = \tfrac{4}{5}x + 2 \]
Verify: test the second point in the finished equation
Why: At x equal to five the equation gives four plus two, which is six — matching the point read off the graph. Testing the point that was not the intercept checks the slope, since the intercept was used to build the equation rather than to test it.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.1 Slope-Intercept Form §5.1, pp. 270-270
Faded example
Use the intercept as one of the two points.
Fill in the blanks
(0, 2) \text2 (5, 6): \quad m = \dfrac0}}___}} = \dfrac______
Why: Both subtractions take the intercept second, which keeps the orders matched. Because its x-coordinate is zero, the denominator is just the other point's x-value, which is why using the intercept as one of the points halves the arithmetic.
Worked example
Any two points on the line work; some are much easier to read.
\[ \text{The line also passes through } (2.5, 4). \text{ Would using that point instead be wise?} \]
Compute with the awkward point
Why: Four minus two over two and a half minus zero.
\[ \frac{2}{2.5} \]
Simplify
Why: Two divided by two and a half is four fifths.
\[ \frac{4}{5} \]
Compare the effort
Why: The same answer, reached through a decimal division.
Draw the lesson
Why: Choose points where the line crosses grid lines exactly.
Figure (svg): Two columns contrasting well-chosen and badly-chosen points for reading a slope
\[ \dfrac{4 - 2}{2.5 - 0} = \dfrac{2}{2.5} = \dfrac{4}{5} \]
Verify: ask what would happen if the point were misread
Why: Reading 2.5 as 2 would give a slope of one rather than four fifths, an error of twenty-five per cent, whereas a lattice point can be read exactly. The risk of a misread is the real reason to prefer grid crossings, not the arithmetic.
Error analysis
The student wrote the equation of a line crossing the axes at (0, 2) and passing through (5, 6).
Annotate
On: \( \begin{aligned} m &= \frac{5 - 0}{6 - 2} = \frac{5}{4} \\ b &= 2 \\ y &= \tfrac{5}{4}x + 2 \end{aligned} \)
Reading a slope back as a sentence catches this reliably: four fifths means four up for every five across, which matches the picture, while five quarters would mean five up for every four across.
Elimination
The line passes through all four of these.
Eliminate the wrong options
Which pair gives the most reliable slope?
Survives elimination: A
Why: Both are exact grid crossings and they are far apart, so neither coordinate is estimated and a small error in either has little effect. Option D is the extreme case worth noticing: two points a tenth of a unit apart make the slope almost entirely a measure of your reading error.
Sorting
One of the two comes free from a graph.
Sort into buckets
Sort each task by whether it needs computation.
The last item in each column is the interesting one: whether counting squares works depends entirely on whether the two axes use the same scale, which the next section takes up.
Socratic
Any two points on the line give the same slope.
Discussion prompt
Explain what is gained by choosing the y-intercept as one of the two points for the slope computation. Then say when you would not be able to.
Hint: Look at the denominator.
Answer:
The intercept has an x-coordinate of zero, so the run is just the other point's x-value and the subtraction in the denominator disappears. It also means one of the two points was read for free, so only one other point has to be found.
You cannot do it when the crossing is off the edge of the drawing, or when it falls between grid lines and cannot be read exactly. Then two other lattice points give the slope, and the intercept is found afterwards — which is precisely the situation Lesson 5.2 is built to handle.
Section
Section 3
Concept
Graphs of real data often use different scales on the two axes. When they do, a square on the grid does not mean one unit in each direction, so the slope must be computed from coordinates rather than counted.
The shuttle graph has 8 units per square across and 4 per square up.
Figure (svg): A graph whose axes use different scales, with the slope computed by formula
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.1 Slope-Intercept Form §5.1, pp. 271-271 — the Study Tip warning that the scales on the axes are different
Picture it
One square across is not one square up.
Figure (svg): A graph whose axes use different scales, with the slope computed by formula
The drawn line looks much steeper than a slope of negative two sevenths would suggest, and that mismatch is entirely an artefact of the scales. The number is right and the picture is misleading.
Worked example
This is Example 3 from the textbook. The scales differ, so the formula is required.
\[ \text{A line passes through } (0, 12) \text{ and } (28, 4) \text{ on a graph with unequal scales. Write its equation.} \]
Read the intercept
Why: The line crosses the vertical axis at (0, 12).
\[ b = 12 \]
Compute the slope by formula
Why: Four minus twelve over twenty-eight minus zero.
\[ -\frac{8}{28} \]
Simplify
Why: Divide both by four.
\[ -\frac{2}{7} \]
Substitute both numbers
Why: Negative two sevenths for m and twelve for b.
\[ y = -(\frac{2}{7}) x + 12 \]
Figure (svg): A descending line modelling a shuttle landing, with both numbers labelled
\[ y = -\tfrac{2}{7}x + 12 \]
Verify: check the second point in the finished equation
Why: At x equal to twenty-eight the equation gives negative eight plus twelve, which is four — matching the point on the graph. That check is independent of the scales, since it works with coordinates rather than with the picture.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.1 Slope-Intercept Form §5.1, pp. 271-271
Prediction
The horizontal axis runs 0 to 40 in steps of 8; the vertical runs 0 to 14 in steps of 4.
Predict first
Can the slope be found by counting squares?
Correct: No, because one square means 8 units across and 4 up.
\[ \text{squares: } -\tfrac{2}{3.5} = -\tfrac{4}{7} \qquad \text{units: } -\tfrac{8}{28} = -\tfrac{2}{7} \]
Why: Counting squares measures a ratio of squares, and the slope is a ratio of units. Those agree only when the two scales are equal, which they are not here — the horizontal square is twice the vertical one, so a count would double the slope. Regularity of the grid is not enough; the two axes must also agree with each other.
Worked example
Seeing the wrong answer makes the warning concrete.
\[ \text{On the shuttle graph the line falls } 2 \text{ squares over } 3.5 \text{ squares. What slope does counting give?} \]
Count the squares
Why: Two down and three and a half across.
\[ -\frac{2}{3.5} \]
Simplify the count
Why: Two over three and a half is four sevenths.
\[ -\frac{4}{7} \]
Compare with the formula
Why: The formula gave negative two sevenths.
Explain the factor
Why: Each vertical square is 4 units and each horizontal square is 8, a ratio of one to two.
Figure (svg): A graph whose axes use different scales, with the slope computed by formula
\[ \text{counting: } -\tfrac{4}{7} \qquad \text{formula: } -\tfrac{2}{7} \]
Verify: check the factor against the two scales
Why: The horizontal scale is twice the vertical one, so counting squares overstates the slope by a factor of two — exactly the discrepancy found. Understanding where the factor comes from is more useful than remembering not to count, because it explains when counting is safe.
Trap
The line falls 2 squares over 3.5 squares on the shuttle graph.
Write the slope as -2/3.5, or about -0.57
Why: Counting squares has worked on every graph so far, where one square meant one unit.
Each square is 4 units up and 8 units across here, so the count measures squares rather than units. The true slope is negative two sevenths, about -0.29.
\[ m = \dfrac{4 - 12}{28 - 0} = -\dfrac{8}{28} = -\dfrac{2}{7} \]
Read the axis labels before counting anything
Why: The formula uses coordinates, which carry their units with them, so it is safe whatever the scales.
A quick glance at the numbers on both axes takes a second and decides whether counting is available.
Faded example
Read the coordinates and subtract.
Fill in the blanks
(0, 12) \text-8 (28, 4): \quad m = \dfrac7___ = \dfrac___}___ = -\dfrac______}
Why: The numerator is negative because the altitude falls, and simplifying negative eight over twenty-eight gives negative two sevenths. Coordinates carry their own units, so this computation is unaffected by how the graph was drawn.
Elimination
A student found a slope of -4/7 for a line whose true slope is -2/7.
Eliminate the wrong options
What is the most likely cause?
Survives elimination: A
Why: A slope exactly double the truth points at a scale ratio of two to one, which is what a square count on this graph produces. Diagnosing an error by the shape of the discrepancy — doubled, reciprocal, or wrong sign — is faster than rechecking every step, and each of the four options here has its own signature.
Socratic
It would be simpler to make one square one unit everywhere.
Discussion prompt
Explain why a graph of real data usually has to use different scales on its two axes. Then say what a reader has to do differently because of it.
Hint: Think about the shuttle's numbers.
Answer:
The shuttle descends through twelve thousand feet over forty thousand feet of approach. Drawing both to the same scale would make the graph either enormous or so flat that nothing could be read from it. Different scales let both ranges fit on a page at a readable size, which is why almost every real graph uses them.
A reader has to check the axis labels before drawing any conclusion about steepness, and must compute slopes from coordinates rather than from the picture. It also means the visual steepness of two graphs cannot be compared unless their scales match — a fact that is regularly exploited to make a modest trend look dramatic.
Section
Section 4
Concept
When a quantity falls steadily, the line modelling it has a negative slope. The intercept is the starting value and the slope is how much is lost per unit of the input.
The shuttle starts at 12,000 feet and loses 2 feet of altitude for every 7 feet of approach.
Figure (svg): A descending line modelling a shuttle landing, with both numbers labelled
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.1 Slope-Intercept Form §5.1, pp. 271-271 — Example 3, Model Negative Slope
Picture it
Twelve thousand feet, falling two for every seven.
Figure (svg): A descending line modelling a shuttle landing, with both numbers labelled
Two sevenths is the shuttle's glide ratio, and it is what decides how far it can reach without power. A larger size of slope would mean a steeper and shorter descent.
Worked example
Reading the two numbers back into the situation.
\[ \text{For } y = -\tfrac{2}{7}x + 12, \text{ altitude in thousands of feet, say what each number means.} \]
Identify the intercept
Why: The constant is twelve.
\[ b = 12 \]
Say what it means
Why: At the start of the approach the altitude is 12,000 feet.
Identify the slope
Why: The coefficient is negative two sevenths.
\[ m = -\frac{2}{7} \]
Say what it means
Why: The shuttle loses 2 feet of altitude for every 7 feet of approach.
Figure (svg): A descending line modelling a shuttle landing, with both numbers labelled
\[ b = 12 \text{ (thousand ft)}, \quad m = -\tfrac{2}{7} \]
Verify: check the units of the two numbers
Why: The intercept is in thousands of feet and the slope is a ratio of feet to feet, which has no units at all. A slope with no units is what makes it a glide ratio rather than a speed, and it is why it can be quoted as two in seven without saying per what.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.1 Slope-Intercept Form §5.1, pp. 271-271
Matching
Each number carries a meaning in its situation.
Match the pairs
Why: In both models the intercept is the starting value and the negative slope is the rate of loss. The shuttle's slope has no units because it compares feet with feet, while the phone's is in dollars per month — a difference worth noticing when reporting an answer.
Worked example
The model answers a question the graph only suggests.
\[ \text{How far along the approach does the shuttle reach } 2000 \text{ feet?} \]
Set the altitude to 2
Why: The units are thousands of feet.
\[ 2 = -(\frac{2}{7}) x + 12 \]
Subtract twelve
Why: Negative ten is left on the left.
\[ -10 = -(\frac{2}{7}) x \]
Multiply by negative seven halves
Why: The coefficient is undone.
\[ x = 35 \]
State the answer
Why: Thirty-five thousand feet along the approach.
Figure (svg): The solution to Worked example use the model to predict shown as a ladder of expressions, one row per algebraic move
\[ x = 35 \text{ (thousand ft)} \]
Verify: check the answer in the original equation
Why: At x equal to thirty-five the model gives negative ten plus twelve, which is two — the altitude asked about. Solving and then substituting back is the routine from Lesson 3.4, applied to a model rather than to a bare equation.
Trap
\[ (0, 12) \text{ and } (28, 4) \]
Compute the slope as 8/28 = 2/7, since the difference is 8
Why: Subtracting the smaller altitude from the larger keeps the arithmetic positive.
\[ y = \tfrac{2}{7}x + 12 \quad \text{(wrong)} \]
That model has the shuttle climbing from 12,000 feet, which is the opposite of what the graph shows and impossible for an unpowered glider.
\[ m = \dfrac{4 - 12}{28 - 0} = -\dfrac{2}{7} \]
Subtract in the same order top and bottom, negative or not
Why: The order is fixed by which point was labelled first, not by which value is larger.
Reading the finished model back as a sentence catches it: a positive slope would say the altitude increases, which contradicts the word descends in the problem.
Faded example
Both numbers, signs included.
Fill in the blanks
\text- 12, \text12 \tfrac______ \text___: \quad y = ___\tfrac______x + ___
Why: The minus sign on the slope is what makes the altitude decrease, and the intercept is the value before the approach begins. Without the sign the model would describe a climb, which is why reading the finished equation back as a sentence is worth doing.
Prediction
Compare a glide ratio of -2/7 with one of -1/2.
Predict first
Which shuttle reaches the ground sooner, and why?
Correct: The one with slope -1/2, since it loses altitude faster per unit along.
\[ \tfrac{2}{7} \approx 0.29 \qquad \tfrac{1}{2} = 0.5 \]
Why: One half is larger than two sevenths in size, so that glider drops more altitude for each unit of forward travel and runs out of height sooner. Comparing negative slopes means comparing their distance from zero, which is the absolute-value idea from Lesson 2.2 — and the second option is the trap of comparing numerators while ignoring denominators.
Socratic
The line eventually reaches the horizontal axis.
Discussion prompt
For the shuttle model, find where the line crosses the horizontal axis and say what that point means. Then say whether the model should be trusted there.
Hint: Set the altitude to zero.
Answer:
Setting y to zero gives negative two sevenths x equals negative twelve, so x is forty-two — the shuttle reaches zero altitude forty-two thousand feet along the approach. That is the touchdown point, and finding it is exactly the x-intercept computation from Lesson 4.4.
Whether to trust it is a separate question. The model was fitted to the descent from twelve thousand to two thousand feet, and the final approach involves flaring the nose up, so the real glide is not linear near the ground. Extending a model past the range it was built for is where correct arithmetic most often produces a wrong answer.
Section
Section 5
Concept
A graph can describe a trend even when no data point lies exactly on the line. The line's slope reports the rate at which the whole collection is changing.
Olympic winning times in the men's hurdles have fallen steadily, and Exercise 40 asks for the equation of that trend.
Figure (svg): A scatter of points with a trend line drawn through them
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.1 Slope-Intercept Form §5.1, pp. 269-276 — the chapter opener on Olympic hurdling times and Exercise 40
Picture it
The line goes near the points rather than through them.
Figure (svg): A scatter of points with a trend line drawn through them
The two points used for the slope must be taken from the drawn line rather than from the data, or the slope will describe two particular observations instead of the trend.
Worked example
The line is drawn; the equation follows as usual.
\[ \text{A trend line crosses the vertical axis at } (0, 7) \text{ and passes through } (8, 3). \]
Read the intercept
Why: Seven, at the crossing.
\[ b = 7 \]
Compute the slope from two points on the line
Why: Three minus seven over eight minus zero.
\[ -\frac{4}{8} \]
Simplify
Why: Negative one half.
\[ m = -\frac{1}{2} \]
Write the equation
Why: Substitute both numbers.
\[ y = -(\frac{1}{2}) x + 7 \]
Figure (svg): A scatter of points with a trend line drawn through them
\[ y = -\tfrac{1}{2}x + 7 \]
Verify: check that the line's points were used rather than the data's
Why: The points (0, 7) and (8, 3) are on the drawn line, and no data point need equal either of them. Using two scattered observations instead would give the slope between those two points rather than the slope of the trend, which can be very different.
Elimination
A trend line is drawn through a scatter of observations.
Eliminate the wrong options
Which pair should be used to compute its slope?
Survives elimination: A
Why: The line has already been positioned to represent the whole collection, so points taken from it inherit that averaging. Option C is worth naming explicitly, because using extremes feels systematic and is one of the least reliable choices available.
Worked example
What the two numbers say about the data.
\[ \text{For } y = -\tfrac{1}{2}x + 7 \text{ modelling a falling measurement over time, interpret both numbers.} \]
Interpret the intercept
Why: Seven is the value at the start of the period.
Interpret the slope
Why: The measurement falls by half a unit each period.
State the limitation
Why: No individual observation need equal the model's prediction.
State the range
Why: The model was fitted between zero and about nine.
Figure (svg): A scatter of points with a trend line drawn through them
\[ b = 7, \quad m = -\tfrac{1}{2} \text{ per period} \]
Verify: compare a prediction against a real observation
Why: At an input of four the model predicts five, and the observed value there was about 5.2. The gap is small and is expected, since a trend line describes the collection rather than any one member of it.
Trap
The observed points include (1, 6.8) and (2, 5.7).
Use those two to compute the slope: -1.1
Why: They are real observations, so they look like the most trustworthy source.
That is the slope between two particular measurements, including whatever noise each carries. The trend's slope is about negative one half, so the estimate is more than double.
Take both points from the drawn line, not from the data
Why: The line already averages out the scatter, which is what it was drawn to do.
Two adjacent observations are the worst possible choice, since they are close together and each carries its own error — the same problem as reading a slope from two nearby points in Lesson 4.4.
Faded example
Read the intercept, compute the slope.
Fill in the blanks
(0, 7) \text2 (8, 3): \quad m = \dfrac7___ = -\dfrac______}, \; b = ___
Why: The slope is negative one half and the intercept seven, giving y equals negative one half x plus seven. Both points came from the drawn line rather than from the scattered data, which is what makes the result a description of the trend.
Hypothesis
Two people draw slightly different lines through the same scatter.
Predict first
What is most likely to happen to their two equations?
Correct: The slopes and intercepts differ slightly, and both descriptions of the trend agree.
A method that always gives the same answer for the same data does exist; it is called least squares, and it is what a calculator's linear regression computes.
Why: Drawing a trend line by eye is a judgement, so two people get slightly different numbers. When the scatter is genuinely close to linear, both lines lie in the same narrow band and both report the same story about the rate of change. Neither is wrong — which is why a trend equation should be quoted as approximate, and why statistics later provides a rule for picking one line rather than leaving it to the eye.
Socratic
The Olympic hurdling times have fallen steadily for decades.
Discussion prompt
If a trend line for winning times has a slope of about negative 0.05 seconds per year, say precisely what that claims and what it does not. Then say what the model predicts about the year 3000 and whether you believe it.
Hint: Distinguish the collection from the individuals.
Answer:
It claims that across the period studied, winning times have fallen by about five hundredths of a second per year on average. It does not claim that any particular Games was five hundredths faster than the last — individual results scatter, and some Games are slower than the one before.
Extended to the year three thousand it would predict a negative time, which is impossible. The model describes a trend within its range and says nothing reliable outside it; human performance must approach some limit rather than improving without bound. Every trend model carries this warning, and the further you extend it the louder the warning gets.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Lesson 4.7 | Lesson 5.1 | |
|---|---|---|
| You are given | an equation | a slope and an intercept, or a graph |
| You produce | the slope and the intercept | an equation |
| The form is used | to read from | to write into |
The same form serves both directions. Chapter 4 read out of it and Chapter 5 writes into it, which is why the form is worth knowing cold.
Pattern
Whether the two numbers come from words, a graph or data, the same five moves cover it.
Step five is the only real check available, and it has to use a point that did not go into the construction — otherwise it confirms nothing.
OpenStax Elementary Algebra 2e, §4.5 Use the Slope-Intercept Form of an Equation of a Line §4.5
Check
Attach each number to its own letter.
Check your understanding
Write the equation of the line with slope -5 and y-intercept 2.
Answer: A
Why: The slope multiplies x and keeps its sign, and the intercept stands alone. Substituting x equal to zero gives two, which confirms the intercept landed in the right place.
Check
Read one number and compute the other.
Check your understanding
A line crosses the y-axis at (0, 3) and passes through (4, 5). What is its equation?
Answer: A
Why: The slope is five minus three over four minus zero, which is two over four, or one half. The intercept is three, read straight off the crossing. Substituting four gives two plus three, which is five, confirming both numbers.
Check
Check the axis scales before counting.
Check your understanding
On a graph where each horizontal square is 10 units and each vertical square is 5, a line falls 3 squares over 2 squares. What is its slope?
Answer: A
Why: Three vertical squares is fifteen units down and two horizontal squares is twenty units across, so the slope is negative fifteen over twenty, which simplifies to negative three quarters.
Real world
Olympic winning times in the men's 110 metre hurdles have fallen steadily. A trend line for the last several decades crosses the vertical axis at about 13.8 seconds in the base year and passes through about 13.0 seconds sixteen years later.
Discussion prompt
Write the equation of that trend, interpret both numbers, and say what the model would predict for a year fifty years after the base year — and whether you would report that prediction.
Hint: Compute the slope from the two points on the line.
Answer:
\[ m = \dfrac{13.0 - 13.8}{16} = -0.05 \quad \text{so } y = -0.05x + 13.8 \]
The intercept of 13.8 seconds is the winning time in the base year, and the slope of negative five hundredths says times have improved by about a twentieth of a second per year on average across the period.
Fifty years on the model predicts 11.3 seconds, well below anything achieved. That is a warning rather than a forecast: the trend was fitted over a few decades and human performance must eventually approach a limit, so the line's continued descent is an artefact of the model's form. Report the trend within its range and say plainly that extrapolating it is not supported by the data.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
A graph's horizontal axis runs in steps of 8 and its vertical axis in steps of 4. A line falls one square for every square across. What is its slope?
Correct: -1/2, since one square is 4 units up and 8 units across.
\[ m = \dfrac{-4}{8} = -\dfrac{1}{2} \]
A square count is safe only when the two axes use the same scale, which is worth checking before every reading.
Why: The slope is a ratio of units and the count is a ratio of squares. One square down is four units and one square across is eight, so the slope is negative four over eight, or negative one half. The first option is the trap and it is the natural instinct, because counting squares has worked on every equally scaled grid so far. The third gets the direction of the correction backwards, doubling instead of halving.
Explain it
They can read a slope and intercept out of an equation and freeze when asked to write one.
Discussion prompt
In no more than four sentences, explain how to write the equation of a line from a graph. Then tell them the one thing to check before computing a slope from a picture.
Hint: Two numbers, one read and one computed.
Answer:
A usable answer: find where the line crosses the vertical axis and write that number down as the constant. Then pick two points where the line crosses grid lines exactly, subtract the y-values for the rise and the x-values for the run in the same order, and that fraction is the coefficient of x. Put the two numbers into y equals mx plus b.
The thing to check first is what one square means on each axis. If the two scales differ, counting squares gives the wrong answer, and you have to read the coordinates and use the formula instead — which is safe either way, so it is the habit worth having.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: The placement is fixed by substituting x equal to zero into the finished equation and checking the intercept comes back. Point choice is fixed by using exact grid crossings that are far apart. Unequal scales are fixed by reading the axis labels first and always using the formula. The sign is fixed by reading the answer back as a sentence and checking it against the direction the line goes. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page write a slope and a y-intercept of your own choosing, with the intercept negative, and build the equation from them showing the brackets before you simplify. Underneath, draw a coordinate plane with equal scales, draw a line on it, mark its crossing on the vertical axis, choose two grid crossings far apart, draw the rise-and-run triangle between them, and write the equation you get. To the right, redraw the same line on a plane whose horizontal scale is twice the vertical, and write beside it both the slope a square count would give and the slope the formula gives, with a note on the factor between them. In the lower half, draw six scattered points that fall from left to right, draw a trend line through them, and write its equation using two points taken from the line rather than from the data. Finally, in the margin, write one sentence saying what the intercept and slope would mean if your trend line described a real falling quantity.
Your two slopes in the third panel should differ by exactly the ratio of the two scales. If they differ by anything else, one of the two readings has an additional error in it.
Recap
Five things, and the third is the one that catches people on real data.
| If the question says | Your first move is |
|---|---|
| The slope is m and the intercept is b | Substitute both into y = mx + b |
| Write the equation of this graph | Read the crossing, then compute the slope |
| The scales on the axes are different | Read coordinates and use the formula |
| The quantity decreases | Expect a negative slope and keep the sign |
| Describe the trend | Take two points from the line, not the data |
Lesson 5.2 handles the case this one cannot: when the y-intercept is not visible or not given. Knowing the slope and any one point turns out to be enough, and the point-slope form is what packages that.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.1 Slope-Intercept Form §5.1, pp. 269-276 — everything on these slides traces back here
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