The slope-intercept form y equals mx plus b, in which the coefficient of x is the slope and the constant is the y-intercept. Includes rewriting an equation into that form, graphing a line from its two numbers without any table, interpreting slope and intercept in a real model, and identifying parallel lines by equal slopes.
Subject: Algebra 1 · 65 slides · symbolic lesson
Open the interactive version of this deck
Title
Algebra 1 · Chapter 4 — Graphing Linear Equations and Functions
Graphing Lines Using 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. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 243-250 — the lesson these objectives are drawn from
Warm-up
Lesson 4.4 read the y-intercept off the constant term. Lesson 4.5 computed a slope from two points. This lesson notices that both numbers were sitting in the equation all along.
Discussion prompt
For y equals 2x plus 3, find the point at x equal to 0 and the point at x equal to 1, then compute the slope between them. Where do both answers appear in the equation?
Hint: Substituting zero and one is deliberately easy.
Answer:
\[ (0, 3) \text{ and } (1, 5) \quad m = \dfrac{5 - 3}{1 - 0} = 2 \]
The slope is two, which is the coefficient of x, and the y-intercept is three, which is the constant term. Both numbers were already written down, so no computation was needed at all — only reading.
Concept
A linear equation written as y equals mx plus b is in slope-intercept form, where m is the slope and b is the y-intercept. Those two numbers determine the line completely.
slope-intercept form — The form y equals mx plus b of a linear equation, in which m is the slope of the line and b is its y-intercept.
Everything needed to draw the graph is visible without any substitution.
Figure (svg): The slope-intercept form with its two parts labelled
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 243-243
Section
Section 1
Concept
Substituting zero into y equals mx plus b gives the point (0, b), so b is the y-intercept. Substituting one gives (1, m plus b), and the slope between those two points is m.
\[ m = \dfrac{(m + b) - b}{1 - 0} = m \]
The run is one because the x-values are zero and one, so the rise is the whole slope.
Figure (svg): Two points on the line y equals 2x plus 3 used to show the slope is 2
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 243-243 — the Slope-Intercept Form box and the derivation beside it
Picture it
Zero and one are the two easiest inputs.
Figure (svg): Two points on the line y equals 2x plus 3 used to show the slope is 2
Notice the run is one, which is what makes the rise equal to the slope rather than merely proportional to it. That is the whole trick of the derivation.
Worked example
The textbook's example is y equals 2x plus 3.
\[ \text{Show that } y = 2x + 3 \text{ has slope } 2 \text{ and } y\text{-intercept } 3. \]
Substitute zero
Why: Two times zero plus three is three, giving the point (0, 3).
\[ b = 3 \]
Name the y-intercept
Why: The point is on the vertical axis, so its y-value is the intercept.
\[ \text{y-intercept } 3 \]
Substitute one
Why: Two times one plus three is five, giving (1, 5).
\[ (1, 5) \]
Compute the slope
Why: Five minus three over one minus zero is two.
\[ m = 2 \]
Figure (svg): Two points on the line y equals 2x plus 3 used to show the slope is 2
\[ m = 2, \quad b = 3 \]
Verify: check with a third point
Why: At x equal to two the equation gives seven, and the slope from (1, 5) to (2, 7) is two again. Any pair of points gives the same slope, as Lesson 4.5 established, so the choice of zero and one was for convenience rather than necessity.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 243-243
Matching
All four are already in slope-intercept form.
Match the pairs
Why: The coefficient of x is the slope and the constant is the intercept, sign included. The last one has no constant written, which means it is zero — that is a direct variation model from Lesson 4.6, and its line passes through the origin.
Worked example
The same two substitutions, with letters instead of numbers.
\[ \text{Show that } y = mx + b \text{ always has slope } m. \]
Substitute zero
Why: m times zero plus b is b, giving the point (0, b).
\[ (0, b) \]
Substitute one
Why: m times one plus b is m plus b, giving (1, m + b).
\[ (1, m + b) \]
Apply the slope formula
Why: The rise is m plus b minus b, and the run is one minus zero.
\[ \frac{m}{1} \]
Simplify
Why: The b terms cancel and the denominator is one.
Figure (svg): The solution to Worked example the general argument shown as a ladder of expressions, one row per algebraic move
\[ m = \dfrac{(m + b) - b}{1 - 0} = \dfrac{m}{1} = m \]
Verify: notice which letter cancelled and which did not
Why: The b cancelled in the numerator, which is why the intercept has no effect on the slope. That is exactly what parallel lines will turn out to be: different values of b with the same m, giving lines that tilt identically and sit at different heights.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 243-243
Trap
\[ 2x - y = 3 \]
Read the coefficient of x as the slope and the constant as the intercept: m = 2, b = 3
Why: Both numbers are visible and in roughly the right places.
The form requires y alone on one side. Rearranging gives y equals 2x minus 3, so the intercept is negative three rather than three.
\[ 2x - y = 3 \;\Longrightarrow\; y = 2x - 3 \quad m = 2, \; b = -3 \]
Rearrange into y equals something before reading anything off
Why: The form is what makes the reading valid; without it the two numbers mean nothing in particular.
Checking one point catches it: at x equal to zero the original gives negative three, not three.
Faded example
The equation is already in the form.
Fill in the blanks
y = -3x + 2: \quad m = -3, \; b = 2
Why: The sign belongs to the coefficient, so the slope is negative three and the line falls. Dropping the minus sign is the commonest slip here, and it reverses the direction of the whole graph.
Elimination
Check whether each is in slope-intercept form first.
Eliminate the wrong options
Which of these lines has slope 3?
Survives elimination: A
Why: Only the first is in slope-intercept form with a coefficient of three on x. The other three each contain the digit three somewhere else, which is exactly the trap: the number's position in the form, not its presence in the equation, is what makes it the slope.
Socratic
The derivation chooses zero and one deliberately.
Discussion prompt
Explain why substituting x equal to zero and x equal to one makes the slope computation come out to m with no arithmetic left over. Then say what would happen if you substituted zero and two instead.
Hint: Look at what the denominator of the slope becomes.
Answer:
The run is one minus zero, which is one, and dividing by one changes nothing. So the slope is simply the rise, which is the amount the equation added when x moved from zero to one — and that amount is m by the structure of the equation.
Substituting zero and two would give the points (0, b) and (2, 2m plus b), so the rise is 2m and the run is 2, and the slope is still m after dividing. The answer is the same, as it must be, but the division is no longer free. Choosing a run of one is what makes the argument read off rather than compute.
Section
Section 2
Concept
Most equations do not arrive in slope-intercept form. Isolating y using the moves from Lesson 3.7 puts them into it, after which the slope and intercept can be read off directly.
Dividing by a negative coefficient changes the sign of every term.
Figure (svg): An equation rearranged step by step into slope-intercept form
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 243-243 — Example 1, Find the Slope and y-Intercept
Picture it
Subtract, divide, then read.
Figure (svg): An equation rearranged step by step into slope-intercept form
The division by negative one is the step that catches people, because it changes the sign of both remaining terms rather than just one.
Worked example
This is Example 1 from the textbook.
\[ \text{Find the slope and } y\text{-intercept of } \; 2x - y = 3. \]
Write the original equation
Why: It is in standard form.
\[ 2 x - y = 3 \]
Subtract 2x from each side
Why: The y-term is left alone on the left.
\[ -y = -2 x + 3 \]
Divide each side by negative one
Why: Every term changes sign.
\[ y = 2 x - 3 \]
Read off both numbers
Why: The coefficient of x is two and the constant is negative three.
\[ m = 2, b = -3 \]
Figure (svg): An equation rearranged step by step into slope-intercept form
\[ y = 2x - 3 \quad m = 2, \; b = -3 \]
Verify: check the intercept against Lesson 4.4
Why: Setting x to zero in the original gives negative y equals three, so y is negative three — the same intercept. Two independent routes to the same number is a real check, and it also confirms the sign change survived the division.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 243-243
Translation
Isolate y, dividing every term.
Match the pairs
Why: Each rearrangement moves the x-term across and then divides every remaining term by the coefficient of y. The second and fourth are the two lines from Example 4 of the textbook, and comparing their slopes of one half and negative one half is what decides whether they are parallel.
Worked example
Two moves rather than one, and every term must be divided.
\[ \text{Rewrite } \; 3x + 2y = 8 \; \text{ in slope-intercept form.} \]
Subtract 3x from each side
Why: The y-term is alone on the left.
\[ 2 y = -3 x + 8 \]
Divide every term by two
Why: Both the negative 3x and the eight are divided.
\[ y = (-\frac{3}{2}) x + 4 \]
Read off m
Why: The coefficient of x is negative three halves.
\[ m = -\frac{3}{2} \]
Read off b
Why: The constant is four.
\[ b = 4 \]
Figure (svg): The solution to Worked example a coefficient on y shown as a ladder of expressions, one row per algebraic move
\[ y = -\tfrac{3}{2}x + 4 \]
Verify: substitute a convenient value into both forms
Why: At x equal to two the rewritten form gives negative three plus four, which is one. In the original, six plus two is eight, which matches. Choosing an x divisible by the denominator keeps the check free of fractions.
Error analysis
The student rewrote 2x minus y equals 3 into slope-intercept form.
Annotate
On: \( \begin{aligned} 2x - y &= 3 \\ -y &= -2x + 3 \\ y &= -2x + 3 \\ m &= -2, \; b = 3 \end{aligned} \)
Whenever a division changes a sign, check both terms on the other side. The error is invisible in the algebra and obvious the moment one point is tested.
Faded example
The x-term has been moved. Divide.
Fill in the blanks
-y = -2x + 3 \;\Longrightarrow\; y = 2x - 3
Why: Dividing by negative one flips the sign of both terms, so negative 2x becomes 2x and positive three becomes negative three. Changing only one of them is the error the check on the intercept catches immediately.
Elimination
The equation is 4x plus 2y equals 10.
Eliminate the wrong options
Which is the slope-intercept form?
Survives elimination: A
Why: Subtracting 4x gives 2y equals negative 4x plus 10, and dividing every term by two gives y equals negative 2x plus 5. Substituting x equal to one gives y equal to three in the answer, and four plus six is ten in the original — a one-line check that rejects all three of the others.
Socratic
Standard form describes the same line.
Discussion prompt
Explain what slope-intercept form makes possible that standard form does not, and say when you would leave an equation in standard form instead.
Hint: Think about what each form lets you see without computing.
Answer:
Slope-intercept form shows the slope and the y-intercept without any computation, so a line can be graphed by reading rather than substituting. It also makes comparing two lines easy, since parallel lines are visible at a glance once both are in the form.
Standard form is worth keeping when you want both intercepts, which Lesson 4.4 read off it in two one-step equations, and it is the form systems of equations are usually written in for Chapter 7. Neither form is better in general — which one to use depends on what the question is asking for.
Section
Section 3
Concept
To graph an equation in slope-intercept form, plot the point (0, b), then use the slope as a pair of moves to reach a second point, and draw the line through them.
A whole-number slope has a denominator of one, so the run is one step right.
Figure (svg): Graphing y equals negative 3x plus 2 by plotting the intercept and using the slope
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 244-244 — Example 2, Graph an Equation in Slope-Intercept Form
Picture it
Down three, right one.
Figure (svg): Graphing y equals negative 3x plus 2 by plotting the intercept and using the slope
No values are substituted and no equations solved. Compare this with the five substitutions a table needed in Lesson 4.2 for the same job.
Worked example
This is Example 2 from the textbook.
\[ \text{Graph the equation } \; y = -3x + 2. \]
Read off m and b
Why: The slope is negative three and the intercept is two.
\[ m = -3, b = 2 \]
Plot the intercept
Why: The point (0, 2) is on the vertical axis.
\[ (0, 2) \]
Write the slope as a fraction
Why: Negative three over one, so move three down and one right.
\[ -\frac{3}{1} \]
Step and draw
Why: From (0, 2) that reaches (1, -1); draw the line through both.
Figure (svg): Graphing y equals negative 3x plus 2 by plotting the intercept and using the slope
\[ \text{through } (0, 2) \text{ and } (1, -1) \]
Verify: check the second point in the equation
Why: Substituting x equal to one gives negative three plus two, which is negative one — matching the point the stepping reached. The step and the substitution agree, which confirms the slope was read with the right sign.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 244-244
Translation
Write each slope as rise over run and read the two moves.
Match the pairs
Why: A whole number has a hidden denominator of one, so its run is a single step right. The negative sign always attaches to the vertical move, because the convention is to read the graph left to right and so to keep the run positive.
Worked example
A fractional slope makes the two moves obvious rather than harder.
\[ \text{Graph } \; y = \tfrac{2}{3}x - 2. \]
Read off m and b
Why: Two thirds and negative two.
\[ m = \frac{2}{3}, b = -2 \]
Plot the intercept
Why: The point (0, -2), two units below the origin.
\[ (0, -2) \]
Read the slope as moves
Why: Up two and right three.
\[ \text{up } 2,\text{ right } 3 \]
Step and draw
Why: From (0, -2) that reaches (3, 0); draw the line.
Figure (svg): A slope written as a fraction and read as a pair of moves
\[ \text{through } (0, -2) \text{ and } (3, 0) \]
Verify: notice which point the step landed on
Why: The second point (3, 0) is on the horizontal axis, so it is the x-intercept from Lesson 4.4. Stepping by a fractional slope often lands on a grid point exactly, which is why fractional slopes are easier to graph this way than to tabulate.
Trap
\[ y = -3x + 2 \quad \text{from } (0, 2) \]
Move 3 up and 1 right, since 3 and 1 are the numbers
Why: The sign is treated as belonging to the whole fraction rather than to the movement.
That reaches (1, 5), which is not on the line — substituting one into the equation gives negative one. The line drawn would rise where it should fall.
\[ m = \dfrac{-3}{1}: \; \text{down } 3, \text{ right } 1 \]
Put the negative sign on the rise and keep the run positive
Why: Moving right is the convention, so the sign has to live in the vertical move.
Substituting the second point into the equation is a one-line check, and it catches a wrong-direction step every time.
Faded example
Start at the intercept and take one step.
Fill in the blanks
y = -3x + 2: \text1 (0, 2) \text-1 (___, ___)
Why: The step lands on (1, -1), and substituting x equal to one into the equation gives negative three plus two, which is negative one. The step and the substitution agree, which is the check worth doing once for every graph.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Method | Points computed | Needs which form |
|---|---|---|
| Table (4.2) | about five | function form |
| Intercepts (4.4) | two | standard form |
| Direct variation (4.6) | one | y = kx |
| Slope-intercept (4.7) | none | y = mx + b |
The last row computes nothing at all: both numbers are read off and the second point is reached by counting squares. That is why this is the method most people end up using.
Socratic
Counting squares feels less rigorous than substituting.
Discussion prompt
Explain why moving up by the rise and right by the run from a point on the line always lands on another point of the line. Then say what would go wrong on a curve.
Hint: Recall what slope measures between any two points.
Answer:
The slope is the ratio of rise to run between any two points on the line, as Lesson 4.5 established. So a move with exactly that ratio, starting from a point on the line, must land on the line — the new point has precisely the relationship to the old one that membership requires.
On a curve the ratio between two points changes depending on which two you pick, so a fixed step would drift off the graph. That is the same reason a curve has no single slope, and it is why this method is specific to straight lines.
Section
Section 4
Concept
In a linear model of a real situation, the y-intercept is the value before anything happens and the slope is the rate at which the quantity changes per unit.
Both numbers should be reported with the units of the situation.
Figure (svg): A production cost model with its intercept and slope interpreted
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 244-244 — Example 3, Use a Linear Model, on hat production costs
Picture it
Three hundred and fifty dollars to start, and a dollar ninety a hat.
Figure (svg): A production cost model with its intercept and slope interpreted
The model came from eight months of data. Reading its two numbers back into the situation is what makes it a model rather than an equation.
Worked example
This is Example 3 from the textbook. Chai makes decorated hats.
\[ \text{Her monthly cost is } y = 1.9x + 350 \text{ for } x \text{ hats. Interpret both numbers.} \]
Identify the y-intercept
Why: The constant term is 350.
\[ b = 350 \]
Say what it means
Why: At zero hats the cost is 350 dollars, so that is her initial cost.
\[ \$ 350\text{ to start} \]
Identify the slope
Why: The coefficient of x is 1.9.
\[ m = 1.9 \]
Say what it means
Why: Each additional hat adds 1 dollar 90 to the cost.
\[ \$ 1.90\text{ per hat} \]
Figure (svg): A production cost model with its intercept and slope interpreted
\[ b = 350 \text{ dollars}, \quad m = 1.90 \text{ dollars per hat} \]
Verify: check the units of each number
Why: The intercept is in dollars and the slope is in dollars per hat, which is why multiplying the slope by a number of hats gives dollars. If the two numbers had the same units, one of them would have been misread.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 244-244
Matching
Read each number back into its situation.
Match the pairs
Why: The intercept always carries the plain unit and the slope always carries a per. The negative slope in the last row is what makes the balance fall rather than rise, and its size is the amount taken each month.
Worked example
The same model, used for two predictions.
\[ \text{Find the cost of } 35 \text{ hats, and of } 60 \text{ hats.} \]
Substitute 35
Why: 1.9 times 35 is 66.5.
\[ 66.5 + 350 \]
Add the initial cost
Why: The total is 416.50.
\[ \$ 416.50 \]
Substitute 60
Why: 1.9 times 60 is 114.
\[ 114 + 350 \]
Add the initial cost
Why: The total is 464.
\[ \$ 464 \]
Figure (svg): A production cost model with its intercept and slope interpreted
\[ y(35) = 416.50 \qquad y(60) = 464 \]
Verify: check the difference against the slope
Why: Twenty-five more hats cost 47.50 more, and 25 times 1.90 is exactly 47.50. The difference between two predictions depends only on the slope, since the initial cost is the same in both — which is a fast way to check a second prediction once the first is known.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 244-244
Trap
\[ y = 1.9x + 350 \]
Say the slope is 1.9 dollars
Why: Money is the obvious unit, since the whole model is about cost.
The slope is a rate, so its units are dollars per hat rather than dollars. Reported as dollars it would suggest the total cost is a dollar ninety.
The slope is one dollar ninety per hat and the intercept is three hundred and fifty dollars.
Give a rate its per, and a starting value its plain unit
Why: The slope multiplies a number of hats, so its units must be dollars per hat for the product to come out in dollars.
Checking the units of the product is a genuine test of whether the model has been read correctly, and it takes a few seconds.
Faded example
The cost model is y equals 1.9x plus 350.
Fill in the blanks
x = 35: \quad y = 1.9(35) + 350 = 66.5 + 350 = 416.5
Why: The variable part comes to 66.50 and the fixed part is 350, giving 416.50 for thirty-five hats. Splitting the answer into a fixed piece and a variable piece is what makes the two numbers of the model visible in the answer.
Prediction
A gym membership is modelled by c equals 25m plus 60.
Predict first
What does the 60 represent?
Correct: A one-off joining fee of 60 dollars.
\[ m = 0: \; c = 60 \qquad m = 1: \; c = 85 \]
Why: The constant is the cost when the number of months is zero, so it is paid before any month has passed — a joining fee. The 25 is the monthly charge, since it is what each extra month adds. Swapping the roles of the two numbers is the standard error, and checking units settles it: 25 is in dollars per month and 60 in dollars.
Socratic
Not every intercept describes something that can happen.
Discussion prompt
Give an example of a linear model whose y-intercept is mathematically fine and physically meaningless. Then say what you should do when that happens.
Hint: Think about a model fitted to data far from zero.
Answer:
A model of an adult's weight against height might have a large negative intercept, since it would describe the weight of a person of zero height. The number is a consequence of fitting a line to data collected between five and six feet, where the line's behaviour at zero was never observed and never mattered.
The right response is to say so rather than to invent an interpretation. State the range over which the model was fitted, use it only inside that range, and note that the intercept is a fitting parameter rather than a measurement. Extending a model beyond the data it came from is the commonest way a correct equation produces a wrong conclusion.
Section
Section 5
Concept
Parallel lines are different lines in the same plane that never intersect. Two non-vertical lines are parallel exactly when they have the same slope and different y-intercepts. Any two vertical lines are parallel.
parallel lines — Different lines in the same plane that never intersect. Two non-vertical lines are parallel if and only if their slopes are equal and their y-intercepts are not.
Equal slopes and equal intercepts would make them the same line rather than two parallel ones.
Figure (svg): Two parallel lines with equal slopes and a third line crossing them
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 245-245 — the Parallel Lines paragraph and Example 4
Picture it
Two slopes match and one does not.
Figure (svg): Two parallel lines with equal slopes and a third line crossing them
The crossing line has a slope of the same size and the opposite sign, which is a common near-miss. Equal size is not equal.
Worked example
This is Example 4 from the textbook. Three lines, one of which is the odd one out.
\[ \text{Which are parallel? } \; a: x - 2y = 6, \quad b: x - 2y = 2, \quad c: x + 2y = 4. \]
Rewrite line a
Why: Isolating y gives y equals one half x minus three.
\[ m = \frac{1}{2} \]
Rewrite line b
Why: y equals one half x minus one.
\[ m = \frac{1}{2} \]
Rewrite line c
Why: y equals negative one half x plus two.
\[ m = -\frac{1}{2} \]
Compare
Why: Lines a and b share a slope and differ in intercept; line c does not.
Figure (svg): Two parallel lines with equal slopes and a third line crossing them
\[ a: \; m = \tfrac{1}{2}, \quad b: \; m = \tfrac{1}{2}, \quad c: \; m = -\tfrac{1}{2} \]
Verify: confirm the intercepts differ
Why: Line a has intercept negative three and line b negative one, so they are genuinely different lines rather than two forms of the same one. Had the intercepts matched as well, the two equations would describe one line, which is not what parallel means.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 245-245
Sorting
Rewrite each and compare the slope with 2.
Sort into buckets
Sort each line by whether it is parallel to y equals 2x plus 1.
The fourth item is the one worth pausing on. Identical equations describe one line, and a line does not count as parallel to itself, which is why the definition insists the intercepts differ.
Worked example
Guided Practice 5. The odd one out has moved.
\[ \text{Which are parallel? } \; a: 3x - 2y = 6, \quad b: 3x + 2y = 6, \quad c: 6x - 4y = 6. \]
Rewrite line a
Why: y equals three halves x minus three.
\[ m = \frac{3}{2} \]
Rewrite line b
Why: y equals negative three halves x plus three.
\[ m = -\frac{3}{2} \]
Rewrite line c
Why: Dividing by negative four gives y equals three halves x minus three halves.
\[ m = \frac{3}{2} \]
Compare
Why: Lines a and c share a slope; line b has the opposite sign.
Figure (svg): The solution to Worked example three more from guided practice shown as a ladder of expressions, one row per algebraic move
\[ a: \; m = \tfrac{3}{2}, \quad c: \; m = \tfrac{3}{2}, \quad b: \; m = -\tfrac{3}{2} \]
Verify: check that a and c are not the same line
Why: Their intercepts are negative three and negative three halves, which differ, so they are two distinct parallel lines. Line c's coefficients are exactly double line a's on the left but not on the right, which is precisely what shifts it without tilting it.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 245-245
Trap
\[ a: 3x - 2y = 6 \quad \text{and} \quad c: 6x - 4y = 6 \]
Say they are not parallel, since their coefficients are different
Why: The equations look different, and the comparison is made before either is rewritten.
Line c's coefficients are double line a's, which is a rescaling that leaves the slope unchanged. Both slopes are three halves, so the lines are parallel after all.
Rewrite both into slope-intercept form before comparing anything
Why: Only the slope decides parallelism, and the slope is not visible in standard form.
Two standard-form equations can look very different and describe parallel or even identical lines. Rewriting is the only reliable comparison.
Elimination
Rewrite both equations in each pair.
Eliminate the wrong options
Which pair of lines is parallel?
Survives elimination: A
Why: Rewriting the second equation gives y equals 3x minus 4, so both slopes are three and the intercepts are negative one and negative four. Option C is the instructive one: equal slopes are necessary and not sufficient, because the intercepts must also differ for there to be two lines at all.
Faded example
Rewrite each into slope-intercept form.
Fill in the blanks
x - 2y = 6 \rightarrow m = \tfrac-no \qquad x + 2y = 4 \rightarrow m = ___\tfrac______ \qquad \text___ ___
Why: The second slope is negative one half, which is the same size as one half and the opposite sign, so the lines are not parallel — they cross. This is the near-miss the textbook builds Example 4 around, and it is worth being alert to whenever a sign changes between two equations.
Socratic
The definition is about intersection and the test is about slope.
Discussion prompt
Explain why two lines with the same slope and different intercepts can never meet. Then say why the rule has to be stated separately for vertical lines.
Hint: Try to solve the two equations together and see what happens.
Answer:
Setting mx plus b equal to mx plus c subtracts to give b equals c, which is false when the intercepts differ. So no value of x satisfies both equations, meaning there is no crossing point — the algebra reports the impossibility as a false statement, exactly as in Lesson 3.9.
Vertical lines have no slope at all, so the rule as stated cannot be applied to them. Two vertical lines have different x-values everywhere and never meet, so they are parallel, and the textbook has to say this separately because undefined slopes cannot be compared for equality. A definition that relies on a number needs an extra clause wherever that number does not exist.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Form | Looks like | Shows you at a glance |
|---|---|---|
| Standard | Ax + By = C | both intercepts, in one step each |
| Slope-intercept | y = mx + b | the slope and the y-intercept |
| Direct variation | y = kx | the constant ratio, and that it passes through the origin |
All three describe lines and each makes a different fact free. Rewriting between them is the skill that lets you pick whichever one the question wants.
Pattern
Whether you are graphing, comparing or interpreting, the same five moves cover it.
For a comparison question, stop after step two: parallelism is decided by the slopes and intercepts alone, with no graphing needed.
OpenStax Elementary Algebra 2e, §4.5 Use the Slope-Intercept Form of an Equation of a Line §4.5
Check
Rewrite before reading anything off.
Check your understanding
What are the slope and y-intercept of 3x minus y equals 5?
Answer: A
Why: Subtracting 3x gives negative y equals negative 3x plus 5, and dividing by negative one gives y equals 3x minus 5. Substituting x equal to zero into the original confirms an intercept of negative five.
Check
Start at the intercept and step.
Check your understanding
Graphing y equals -2x plus 5 from (0, 5), where does one step by the slope land?
Answer: A
Why: The slope is negative two over one, so the step is two down and one right, reaching (1, 3). Substituting x equal to one into the equation gives negative two plus five, which is three, confirming it.
Check
Compare the slopes after rewriting.
Check your understanding
Which line is parallel to y equals 4x minus 1?
Answer: A
Why: Rewriting gives y equals 4x minus 3, so the slope is four and the intercept is negative three rather than negative one. Equal slope and different intercept is exactly the condition for parallel lines.
Real world
Two phone plans: the first charges 30 dollars a month with no joining fee, the second 20 dollars a month plus a 90 dollar handset.
Discussion prompt
Write each as a linear model, say what the slope and intercept mean, and use the two numbers to say which plan is cheaper and when that changes.
Hint: Write both in slope-intercept form and compare the two numbers separately.
Answer:
\[ c_1 = 30m \qquad c_2 = 20m + 90 \]
The first has slope thirty dollars a month and intercept zero, so it is a direct variation model from Lesson 4.6. The second has slope twenty dollars a month and intercept ninety dollars, the handset paid up front.
The second plan starts ninety dollars behind and gains ten dollars a month, so it catches up after nine months and is cheaper from then on. Neither number alone answers the question: the intercept says who starts ahead and the slope says who is gaining, and the crossing point is where the two effects balance. Chapter 7 will find that crossing point by solving the two equations together.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
What is the slope of the line 4x plus y equals 7?
Correct: -4, once the equation is rewritten as y = -4x + 7.
\[ 4x + y = 7 \;\Longrightarrow\; y = -4x + 7 \quad m = -4, \; b = 7 \]
Why: The slope can only be read off after y has been isolated. Subtracting 4x from both sides gives y equals negative 4x plus 7, so the slope is negative four and the line falls. Reading the coefficient straight off the standard form is the trap, and it produces exactly the wrong sign — the line would be drawn rising when it in fact falls. Substituting two points settles it: at x equal to zero, y is seven, and at x equal to one, y is three.
Explain it
They graph every line by building a five-row table and find it slow.
Discussion prompt
In no more than four sentences, show them how to graph a line from y equals mx plus b without computing anything. Then tell them the one check worth doing afterwards.
Hint: Two numbers, one point, one step.
Answer:
A usable answer: the constant tells you where the line crosses the vertical axis, so plot that point first. The coefficient of x tells you how to step to the next point — write it as a fraction, go up by the top and right by the bottom, and a negative sign means go down instead. Two points, and you are done.
The check is to substitute the second point back into the equation. If it satisfies it, the slope was read with the right sign and stepped in the right direction — which is the only thing that usually goes wrong.
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: Rewriting is fixed by dividing every term and then testing x equal to zero against the original. Stepping is fixed by putting the negative sign on the vertical move and checking the second point in the equation. Interpretation is fixed by attaching units: the intercept gets a plain unit and the slope gets a per. Parallelism is fixed by rewriting both equations before comparing, and by remembering that the intercepts must differ too. 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
Write one equation in standard form at the top of a page and rewrite it into slope-intercept form, showing every move and circling the two numbers at the end. Beside it, draw a coordinate plane, plot the y-intercept, draw the rise-and-run step from it as a dotted right triangle labelled with the two moves, and draw the line. Substitute your second point back into the original equation and write the check underneath. In the lower half, write a second equation with the same slope and a different intercept, graph it on the same plane, and write one sentence saying why the two lines never meet. Finally, in the margin, invent a real situation your first equation could model, and write what the slope and the intercept mean in it, with units on both.
Your two lines should look identically tilted and sit at different heights. If they converge anywhere on the page, one of the two slopes was misread — most likely a sign lost during a division.
Recap
Five things, and the second is the one that makes every graphing question quick.
| If the question says | Your first move is |
|---|---|
| Find the slope and y-intercept | Isolate y, then read both off |
| Graph the equation | Plot (0, b) and step by the slope |
| What does the slope mean here | Give it units of something per something |
| Are these lines parallel | Rewrite both, then compare m and b |
| Which coefficient is the slope | Only the one on x, once y is alone |
Lesson 4.8 closes the chapter by returning to functions. It gives the vertical line test a formal statement, introduces the notation f of x, and separates the functions from the wider class of relations.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.7 Graphing Lines Using Slope-Intercept Form §4.7, pp. 243-250 — everything on these slides traces back here
Want this taught 1-on-1? Alexander tutors Algebra 1 — $55/session, free consultation.