Chapter 5 of Algebra 1: Concepts and Skills, built for a visual learner. Writing a line from its slope and intercept, point-slope form derived from the slope formula, equations from two points, converting to standard form, modelling real situations where the intercept is the start and the slope is the rate, and parallel and perpendicular slopes.
Subject: Algebra 1 · 60 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
Algebra 1 · Chapter 5
Going backwards: from a picture, a pair of points, or a real situation, to the equation itself
Objectives
Chapter 4 went from equation to picture. This chapter runs the other way, which is the harder and far more useful direction.
Figure (svg): Two recipe cards, one holding the slope and one holding a point, combining into a single line equation
Section
Section 5.1
Concept
If you know how steep a line is and where it crosses the vertical axis, the equation writes itself.
\[ y = mx + b \]
Figure (svg): Two recipe cards, one holding the slope and one holding a point, combining into a single line equation
Worked example
A line has slope 3 and crosses the vertical axis at negative 4. Write its equation.
Identify which number is m and which is b
Why: The steepness is m and the crossing point is b. Naming them before substituting is what prevents them being swapped.
Substitute both into the form
Why: Nothing else is needed; the form is already solved for y.
\[ y = 3x - 4 \]
Figure (svg): A line of slope three crossing the vertical axis at minus four, with a staircase step drawn
Verify: test the point one unit across
Why: At x equal to 1 the equation gives 3 minus 4, which is negative 1, and the staircase step also lands at negative 1. The picture and the equation agree.
Prediction
A line crosses the vertical axis at 2 and falls one unit for every one unit across.
Predict first
What is its equation?
Correct: y = -x + 2
Why: Falling one unit per unit across is a slope of negative one, and crossing at 2 makes b equal to 2. Writing the slope as positive one is the commonest slip: falling means the rise is negative even though the number one looks positive.
Error analysis
A student was told the slope is 5 and the y-intercept is 2. Find the error.
Annotate
On: \( y \;\overset{?}{=}\; 2x + 5 \)
Say the form out loud as slope times x, then intercept and the swap becomes hard to make.
Matching
Read the two numbers out of each sentence.
Match the pairs
Why: The last two are the pair worth comparing. A slope of negative one with intercept 4 still has an x in it and slopes downward, while a flat line at 4 has no x term at all because its slope is zero.
Explain it to yourself
The constant in the equation is always where the line meets the vertical axis. Say why.
Discussion prompt
Using the equation y equals m x plus b, explain why b must be the y-intercept.
Hint: What is x equal to everywhere on the vertical axis?
Answer:
The y-axis is where x equals zero. Substituting zero for x gives m times zero, which is nothing, plus b — so y equals b there.
The slope has no effect at that one point, because it is always multiplied by x. That is why b is visible in the equation without any work: it is the one place the slope switches off.
Tweak it
The steepness is locked. Only the starting height is changing.
Parameter explorer
Drag b. What happens to the line, and how do all these lines relate to each other?
\[ y = 2x + {b} \]
Ranking
Read the slope out of each equation, then order them from flattest to steepest.
Put in order
Why: Steepness is the size of the slope regardless of sign, so the order is 0, then a half, then 2, then 3, then 7. The line y equals 9 has no x term at all, making its slope zero and its graph perfectly flat, which puts it first every time.
Section
Section 5.2
Concept
Often you know the slope and some point that is nowhere near the axis. Point-slope form is built for exactly that.
Figure (svg): The point-slope formula with each part labelled: the known point, the slope and the general point
\[ y - y_1 = m(x - x_1) \]
Worked example
Write the equation of the line with slope 2 passing through the point (3, 5).
Label the known point
Why: The known x is 3 and the known y is 5. These are the ones that carry subscripts in the form.
Substitute into point-slope form
Why: Put the numbers in and leave x and y as they are — they stay variable because the equation must describe every point on the line.
\[ y - 5 = 2(x - 3) \]
Rearrange into slope-intercept form
Why: Distribute, then add five to both sides.
\[ y - 5 = 2x - 6 \;\Longrightarrow\; y = 2x - 1 \]
Figure (svg): A line of slope two through the point three comma five, with its y-intercept at minus one marked
Verify: substitute the original point into the final equation
Why: Two times three is six, minus one is five, which is the y coordinate we were given. The rearranged equation still passes through the original point.
Fill the middle
A line has slope negative 3 and passes through (-2, 4). Fill the blanks.
Fill in the blanks
y - 4 = -3(x - (-2)) \;\Longrightarrow\; y = -3x - 2
Why: The known x is negative two, so the form contains x minus negative two, which simplifies to x plus two. Distributing gives negative 3x minus 6, and adding 4 to both sides leaves negative 3x minus 2. The double negative in the x slot is the single most common error here.
Notation
The subscripts in point-slope form confuse people. Decode what each symbol is doing.
Annotate
On: \( y - y_1 = m(x - x_1) \)
Point-slope form is not a new idea. It is the slope formula with the denominator multiplied out.
Socratic
One question, no computation.
\[ m = \frac{y - y_1}{x - x_1} \]
Discussion prompt
Starting from the slope formula above, what single algebraic move produces point-slope form?
Hint: What do you always do to an equation with a fraction in it?
Answer:
Multiply both sides by the denominator. That clears the fraction and gives y minus y-one equals m times the quantity x minus x-one.
So point-slope form is not something extra to memorise — it is the slope formula with the fraction cleared, which is the same move you used all through Chapter 3.
Discrimination
Do not write any equations. Just choose the tool.
Sort into buckets
Sort each situation by the form that gets there fastest.
Faded example
A line has slope 5 and passes through (1, -2). Fill in every blank.
Fill in the blanks
y - (-2) = 5(x - 1) \;\Longrightarrow\; y = 5x - 7
Why: The known y is negative two, so the left side becomes y plus 2. Distributing gives 5x minus 5, and subtracting 2 from both sides leaves 5x minus 7. Checking with the original point: 5 times 1 minus 7 is negative 2, which is correct.
Real world
A tree was 3 metres tall when it was 4 years old, and it grows 0.5 metres a year.
Discussion prompt
Why is point-slope form the right tool here, and what does the y-intercept turn out to mean?
Hint: What height does the model give at time zero?
Answer:
You are given a point (4, 3) and a slope of 0.5, but that point is not the starting height — so slope-intercept form has nothing to substitute yet.
\[ h - 3 = 0.5(t - 4) \;\Longrightarrow\; h = 0.5t + 1 \]
The intercept turns out to be 1 metre: the model says the tree was one metre tall when it was planted. That is a real, checkable prediction that fell out of the algebra rather than being given.
Section
Section 5.3
Concept
Two points determine a line completely. Compute the slope from them, then feed either point into point-slope form.
Figure (svg): Two points on a plane with the slope triangle between them and the resulting line drawn through both
Using either point gives the same final equation, which is a free check on your work.
Worked example
Find the equation of the line through the points below.
\[ (1, 2) \text{ and } (5, 8) \]
Compute the slope
Why: Subtract in a consistent order: second minus first, for both coordinates.
\[ m = \frac{8 - 2}{5 - 1} = \frac{6}{4} = \frac{3}{2} \]
Substitute the slope and one point into point-slope form
Why: Either point works. Choosing the one with smaller numbers keeps the arithmetic cleaner.
\[ y - 2 = \tfrac{3}{2}(x - 1) \]
Rearrange into slope-intercept form
Why: Distribute the fraction, then add two to both sides.
\[ y = \tfrac{3}{2}x + \tfrac{1}{2} \]
Figure (svg): The same two points with the finished line drawn through them and its y-intercept marked at one half
Verify: substitute the other point into the final equation
Why: Three halves of five is seven point five, plus one half is eight, which is the second point's y coordinate. Both given points satisfy the equation.
Hypothesis
You used the point (1, 2). Suppose you had used (5, 8) instead.
Predict first
Would starting from the other point give a different final equation?
Correct: No — the same equation comes out either way.
\[ y - 8 = \tfrac{3}{2}(x - 5) \;\Longrightarrow\; y = \tfrac{3}{2}x + \tfrac{1}{2} \]
Why: Both points lie on the same line, and point-slope form describes the whole line rather than the point you fed it. Starting from (5, 8) gives y minus 8 equals three halves times the quantity x minus 5, which rearranges to exactly the same slope-intercept form. Doing it both ways is a genuinely free check.
Error analysis
A student computed the slope through (2, 9) and (6, 1) like this. Find the fault.
Annotate
On: \( m \;\overset{?}{=}\; \frac{9 - 1}{6 - 2} = \frac{8}{4} = 2 \)
Whenever a slope comes out positive for a line you know is falling, suspect the subtraction order.
Reverse engineer
Work backwards from the finished slope.
Fill in the blanks
m = \frac33} = 2 \;\text___\; ___
Why: The rise is 7 minus 1, which is 6, and the slope is 2, so the run must be 3 because 6 divided by 3 is 2. Reading a slope backwards like this is exactly what you do when a question gives you a slope and asks for a missing coordinate.
Explain it
A classmate asks why you never need three points to write a line's equation.
Discussion prompt
Explain in two sentences why two points fix a line completely, and what a third point would add.
Hint: What are the two ingredients any line equation needs?
Answer:
Two points give you the two things a line needs: the slope comes from the pair, and either point pins down which of the parallel family it is.
A third point adds no new information if it is on the line, and proves one of your points is wrong if it is not. So a third point is a check, never a requirement.
Pattern
The same journey, one stage per frame. Predict the next line before advancing.
Step through it
After the slope is found, what is the very next thing that happens?
Four stages, and the same four every time. The only thing that changes between problems is the arithmetic.
Picture it
A picture of why using either point gives the same line.
Figure (svg): One line drawn through two points, with a slope triangle from each point, both triangles having the same shape
Because the triangles are identical, the two starting points describe the same relationship. Getting different answers means an arithmetic slip, not a genuine ambiguity.
Pattern
Every question in this chapter is a variation on these five moves.
The check is not optional. It catches sign errors, swapped coordinates and distribution slips all at once.
Section
Section 5.4
Concept
Standard form puts both variables on the same side. It is the fastest form for finding intercepts and the natural form for systems in Chapter 7.
Figure (svg): The three common forms of a line equation shown side by side with what each is best for
\[ Ax + By = C \]
Worked example
Write the equation below in standard form with integer coefficients.
\[ y = \tfrac{2}{3}x - 4 \]
Clear the fraction by multiplying every term by 3
Why: Standard form conventionally uses whole numbers, so the denominator goes first.
\[ 3y = 2x - 12 \]
Move the x term to the left side
Why: Subtract 2x from both sides so both variables sit together on the left.
\[ -2x + 3y = -12 \]
Multiply through by negative one so the leading coefficient is positive
Why: This is a convention rather than a requirement, but it is the form answers are usually given in.
\[ 2x - 3y = 12 \]
Figure (svg): The same line drawn once, with both its slope-intercept and standard form equations labelled beside it
Verify: test the y-intercept in the standard form
Why: Substituting x equal to 0 and y equal to negative 4 gives 0 minus negative 12, which is 12, matching the right side. The conversion preserved the line.
Comparison
Fill the blanks. Choosing the right form saves more time than any algebraic trick.
Comparison matrix
| you are asked for | best form | why |
|---|---|---|
| the slope, immediately | slope-intercept | m is sitting right there |
| both intercepts | standard | set each variable to zero in turn |
| an equation from a point and a slope | point-slope | it takes exactly those two inputs |
None of these forms is more correct than the others. They are tools, and the skill is picking one before starting.
Sorting
Sort by form. Watch for the ones that are almost but not quite in a named form.
Sort into buckets
Which form is each written in?
Trap
Convert the equation to standard form.
\[ y = \tfrac{1}{2}x + 3 \]
Multiply the fraction away, but only where the fraction is
Why: The half is the problem, so it feels like the only term that needs multiplying.
\[ y = x + 3 \;\to\; -x + y = 3 \]
Testing x equal to 2: the original gives 4, but this version gives 5. The line has moved.
Convert the same equation by multiplying every term on both sides.
Multiply all three terms by 2, then rearrange
Why: The equals sign is only preserved if both sides are scaled identically, and that means every term.
\[ 2y = x + 6 \;\Longrightarrow\; -x + 2y = 6 \;\Longrightarrow\; x - 2y = -6 \]
Testing x equal to 2: the original gives 4, and this gives 2 minus 8, which is negative 6, matching the right side.
Notation
Standard form hides its slope. Decode what each letter is doing.
Annotate
On: \( Ax + By = C \)
This form trades away the visible slope in exchange for two intercepts that fall out in one line each.
Elimination
Every option is a correct equation of some line. Only one obeys the standard-form convention.
Eliminate the wrong options
Which equation is properly written in standard form with integer coefficients?
Survives elimination: A
Why: Standard form needs both variable terms on the left, a plain constant on the right, and whole-number coefficients. Only the first option satisfies all three at once. The others are all perfectly valid equations describing lines, which is exactly why the convention has to be stated rather than guessed.
Section
Section 5.5
Concept
In a real story the two numbers of a line stop being abstract: b is where you begin and m is how fast it changes.
Figure (svg): A real-world line showing a starting value on the vertical axis and a steady rate of climb
Every linear model question is really asking you to find those two numbers in the words.
Worked example
A pool contains 200 litres and is draining at 15 litres per minute. Write a model and say when it empties.
Identify the starting amount
Why: At time zero there are 200 litres, so the intercept is 200.
Identify the rate and its sign
Why: The pool is losing water, so the rate is negative fifteen litres per minute. Getting this sign right is most of the work.
\[ V = -15t + 200 \]
To find when it empties, set the volume to zero
Why: Emptying means no water left, which is the horizontal-axis crossing.
\[ 0 = -15t + 200 \;\Longrightarrow\; t = \frac{200}{15} \approx 13.3 \]
Figure (svg): A falling line starting at two hundred litres and reaching zero at about thirteen minutes
Verify: check the volume at 10 minutes against common sense
Why: The model gives 200 minus 150, which is 50 litres. Losing 15 litres a minute for 10 minutes should remove 150 litres from 200, leaving 50 — the model and the story agree.
Real world
A gym membership is modelled by the equation below, where C is cost in dollars and m is months.
\[ C = 35m + 90 \]
Discussion prompt
Say in plain English what the 35 and the 90 each mean, and what the model predicts for one year.
Hint: What do you pay if you join and immediately cancel?
Answer:
The 35 is the monthly fee — the rate at which cost grows per month.
The 90 is a one-off joining fee, paid before any months at all. It is what you owe at m equal to zero.
For a year, substitute m equal to 12: 35 times 12 is 420, plus 90 gives 510 dollars.
Interpreting is the real skill here. A number with no meaning attached cannot be checked against common sense.
Missing information
A problem reads: a candle burns down at a steady rate. Write an equation for its height.
Discussion prompt
What are the two facts you must be told before this can be written, and which part of the equation does each supply?
Hint: How many ingredients does a line always need?
Answer:
You need the starting height, which supplies the intercept, and the burn rate with its units, which supplies the slope.
Alternatively two observations at known times would do, because from two points you can compute both. Either way you need two independent facts — one is never enough, because slope alone describes a whole family of parallel lines.
Estimation
Using the pool model, water is draining at 15 litres per minute from 200 litres.
Predict first
Roughly how much water is left after 8 minutes?
Correct: About 80 litres — the exact value is 80.
\[ V = -15(8) + 200 = 80 \]
Why: Fifteen litres a minute for 8 minutes removes about 120 litres, and 200 minus 120 is 80. An answer above 200 would mean the pool was filling rather than draining, which is the sign error this estimate is designed to catch.
Trade off
Plan A costs 60 dollars up front then 10 a month. Plan B costs nothing up front but 25 a month. Fill the blanks.
Comparison matrix
| plan A | plan B | |
|---|---|---|
| equation | C = 10m + 60 | C = 25m |
| cost after 4 months | 100 dollars | 100 dollars |
| better for a long stay | yes, it grows more slowly | no, its steeper slope overtakes |
The two lines cross at 4 months, and which plan wins depends entirely on which side of that crossing you live on. That crossing point is what Chapter 7 is about.
Step zero
A phone battery starts at 100 percent and drops 8 percent per hour of video.
Discussion prompt
Before writing anything: which quantity is the input, which is the output, what is the intercept, and what is the sign of the slope?
Hint: Is the quantity growing or shrinking as time passes?
Answer:
Input: hours of video. Output: battery percentage. Intercept: 100, the charge at zero hours. Slope: negative 8, because the battery is falling.
\[ B = -8h + 100 \]
Deciding the sign before writing is the whole discipline here. A model that predicts a rising battery is wrong in a way no amount of careful arithmetic will fix.
Section
Section 5.6
Concept
Parallel lines have the same slope. Perpendicular lines have slopes that multiply to negative one.
Figure (svg): Two parallel lines with equal slopes beside two perpendicular lines whose slopes multiply to negative one
\[ m_1 \cdot m_2 = -1 \quad \Longleftrightarrow \quad m_2 = -\frac{1}{m_1} \]
Worked example
Write the equation of the line perpendicular to y equals 2x plus 1 that passes through (4, 3).
Read the original slope
Why: The coefficient of x is 2, so the original slope is 2.
Flip it over and change its sign
Why: The reciprocal of 2 is one half, and negating gives negative one half. Both moves are needed — doing only one is the classic error.
\[ m = -\tfrac{1}{2} \]
Substitute the new slope and the given point into point-slope form
Why: The point is on the new line, not the old one.
\[ y - 3 = -\tfrac{1}{2}(x - 4) \]
\[ y = -\tfrac{1}{2}x + 5 \]
Figure (svg): Two lines crossing at a right angle at the point four comma three, one of slope two and one of slope minus one half
Verify: multiply the two slopes together
Why: Two times negative one half is negative one, which is the perpendicularity condition, so the two lines really do meet at a right angle.
Prediction
Commit before computing.
\[ y = -\tfrac{3}{4}x + 2 \]
Predict first
What is the slope of any line perpendicular to this one?
Correct: 4/3
\[ -\tfrac{3}{4} \cdot \tfrac{4}{3} = -1 \]
Why: Flip three quarters over to get four thirds, then change the sign. The original slope is already negative, so negating it makes the perpendicular slope positive. Answering negative four thirds means the flip was done but the sign change was not.
Sorting
Each pair of slopes describes two lines. Sort them.
Sort into buckets
What is the relationship between each pair of lines?
Two truths and a lie
Three claims about parallel and perpendicular lines.
Eliminate the wrong options
Which statement is false?
Survives elimination: C
Why: Keep the false statement, which is C. Negating alone is not enough — the slope must also be flipped over. Slopes of 3 and negative 3 are negatives but not reciprocals, and their product is negative 9 rather than negative 1, so those lines cross at an angle that is nowhere near a right angle.
Analogy
Perpendicular slopes look arbitrary until you connect them to turning.
Match the pairs
Why: Turning a right angle swaps the roles of rise and run and reverses one of them. That is literally what taking the negative reciprocal does to a fraction, which is why the rule is two moves rather than one.
Edge cases
The slope-product rule breaks down in one case. Find out where.
Discussion prompt
A horizontal and a vertical line clearly meet at a right angle. Why can you not verify that by multiplying their slopes?
Hint: What is the slope of a vertical line?
Answer:
The horizontal line has slope zero and the vertical line has no slope at all — it is undefined, not a number. So there is nothing to multiply, and the product rule simply does not apply.
The lines are still perpendicular; it is the algebraic test that fails, not the geometry. This is a good habit to carry forward: a rule stated in terms of numbers stops working the moment one of those numbers does not exist.
\[ m = 0 \quad \text{and} \quad m \text{ undefined} \]
Check
Solve it on paper before you click.
Check your understanding
What is the equation of the line through (2, -1) and (6, 7)?
Answer: A
Why: The slope is 7 minus negative 1 over 6 minus 2, which is 8 over 4, or 2. Using the point (2, -1) in point-slope form gives y plus 1 equals 2 times the quantity x minus 2, which rearranges to y equals 2x minus 5. Checking the second point: 2 times 6 minus 5 is 7.
Check
Solve it on paper before you click.
Check your understanding
Which line is perpendicular to 2x + 3y = 6?
Answer: A
Why: Rearranging gives y equals negative two thirds x plus 2, so the original slope is negative two thirds. Flipping and negating gives positive three halves, and negative two thirds times three halves is negative one.
Commit first
Answer, then rate your confidence.
\[ \text{A line parallel to } y = 4x - 7 \text{ through } (0, 2) \]
Predict first
What is its equation?
Correct: y = 4x + 2
\[ y = 4x + 2 \]
Why: Parallel means the slope is unchanged at 4, and the given point has x equal to zero so it is the y-intercept itself, making b equal to 2. Choosing the negative reciprocal would give a perpendicular line instead, and reusing the original intercept would give the very same line rather than a parallel one.
Exit ticket
Last commitment of the chapter.
Predict first
Which of these is shakiest right now?
Correct: Whatever you picked is the one to drill first.
Why: All four appear together in Chapter 7 when systems of equations arrive, and a wobble in any one of them turns a solvable system into a guessing game. Ten targeted problems now is worth an hour of mixed revision later.
Connect it up
One page, drawn by you.
Draw it
Draw the three forms of a line across the page: slope-intercept, point-slope, standard. Between them draw arrows showing how to convert each way, and label each arrow with the actual move. Off to the side, attach: two points, parallel, perpendicular, real-world model.
If your map has no arrow from two points into point-slope form, add it — that is the single most-used route in the chapter.
Recap
You can now go from a picture, a pair of points, or a sentence about the real world, straight to an equation.
| if you remember one thing | it should be |
|---|---|
| about ingredients | a line needs a slope and a point — always exactly two facts |
| about point-slope | it is just the slope formula with the fraction cleared |
| about forms | converting changes the spelling, never the line |
| about perpendicular | flip it over AND change the sign, not one or the other |
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