C. Algebra · Linear and Non-Linear Relations

The Equation y = mx + b: Slope and y-Intercept

C4.4 C3.2 A1About 70 minutes

Learning goals

I can identify the slope and y-intercept of a line from its equation in the form y = mx + b.
I can connect the slope to the rate of change and the y-intercept to the initial value.
I can graph a line from its equation by starting at the y-intercept and using the slope.
I can rearrange simple equations into the form y = mx + b, and find x- and y-intercepts.

Words to know

y-intercept
The y-coordinate of the point where a line crosses the y-axis (where x = 0). In y = mx + b it is b.

x-intercept
The x-coordinate of the point where a line crosses the x-axis (where y = 0).

slope–y-intercept form
The form y = mx + b of a linear equation, where m is the slope and b is the y-intercept. Ontario's curriculum also writes it as y = ax + b.

parallel lines
Lines in the same plane that never meet. Non-vertical parallel lines have equal slopes.

Putting it together

In the last lessons you found two key numbers for every linear relation:

the rate of change, which on a graph is the slope, and
the initial value, which on a graph is where the line meets the y-axis, the y-intercept.

Every non-vertical line can be written using just these two numbers:

y = mx + b

where m is the slope (rate of change) and b is the y-intercept (initial value). You may also see this written y = ax + b: the letter does not matter, the meaning is the same.

Equation	Slope m	y-intercept b	Meaning
y = 3x + 2	3	2	Starts at 2, goes up 3 for each 1 right
y = −2x + 5	−2	5	Starts at 5, goes down 2 for each 1 right
y = 4x	4	0	Direct variation: passes through the origin
y = 6 − x	−1	6	Rewrite as y = −x + 6
y = 7	0	7	Horizontal line
C = 0.15t + 20	0.15	20	A phone plan: $20 per month plus $0.15 per minute

Read m and b straight from the equation. Watch the sign that comes before each number.

Graphing a line from y = mx + b

Plot the y-intercept (0, b).
Use the slope as rise ÷ run to find more points. For m = 2, think 2/1: right 1, up 2. For m = −2/3: right 3, down 2.
Plot at least three points (the third one is a check), then draw a straight line through them with a ruler, with arrows at both ends.

You can always check a point by substituting it: if (3, 3) is on y = 2x − 3, then 2(3) − 3 should equal 3. It does.

Left: start at the y-intercept −3, then step right 1, up 2 again and again. Right: start at 4, then step right 3, down 2. Each dashed step is one slope triangle.

Families of lines

What happens when you change just one number?

Keep b and change m: the lines all pass through the same point on the y-axis but tilt differently. A bigger m makes a steeper line; a negative m makes the line fall.
Keep m and change b: the lines all have the same steepness, so they are parallel. Changing b slides the line up or down.

Left: four lines through (0, 1) with different slopes; the dashed line has a negative slope. Right: three lines with slope 1. They are parallel; b moves each line up or down.

Worked example

Example 1: Graph y = −(2/3)x + 4 and find the intercepts

Graph y = −(2/3)x + 4. State the y-intercept and the x-intercept.

b = 4, so plot (0, 4). That is the y-intercept.
m = −2/3: from (0, 4) go right 3 and down 2 to (3, 2). Again: right 3, down 2 to (6, 0).
Draw the line through the three points.
The x-intercept is where y = 0. From the graph it is (6, 0). Check algebraically: 0 = −(2/3)x + 4, so (2/3)x = 4 and x = 4 × 3/2 = 6 ✓.

y-intercept 4, x-intercept 6.

Worked example

Example 2: Rearrange into y = mx + b

Find the slope and y-intercept of 3x + y = 7 and of 2y = 4x − 10.

3x + y = 7: subtract 3x from both sides: y = −3x + 7. So m = −3, b = 7.
2y = 4x − 10: divide every term by 2: y = 2x − 5. So m = 2, b = −5.

3x + y = 7 → y = −3x + 7 (m = −3, b = 7). 2y = 4x − 10 → y = 2x − 5 (m = 2, b = −5).

Worked example

Example 3: Meaning in context

A phone plan costs C = 0.15t + 20 dollars per month, where t is the number of minutes of long-distance calls. Explain the slope and y-intercept, and find the cost for 120 minutes.

Slope m = 0.15: the cost goes up $0.15 for every minute. This is the rate of change.
y-intercept b = 20: with 0 minutes, the bill is $20. This is the fixed monthly fee (initial value).
C = 0.15(120) + 20 = 18 + 20 = 38.

$0.15 per minute plus a $20 monthly fee. 120 minutes cost $38.

Common mistakes

In y = 6 − x, the slope is −1, not 6. Rewrite it as y = −x + 6 first.
When you rearrange, divide every term: 2y = 4x − 10 becomes y = 2x − 5, not y = 2x − 10.
A negative slope like −2/3 means right 3, down 2 (or left 3, up 2). Do not make both parts negative.

Just for fun

What did y = 2x say to y = 2x + 3?

"We have so much in common, but I don't think we'll ever meet."

Hands-on activity

Card-deck lines

You need: a deck of playing cards with the face cards and 10s removed, graph paper (or a printed grid), a ruler, a pencil.

Black cards are positive and red cards are negative. An ace is 1.
Draw one card for m and one card for b. Write the equation y = mx + b.
Graph it: plot b on the y-axis, then use the slope to plot two more points. Draw the line.
Check one point by substituting it into the equation.
Do this four times on the same grid. Did any of your lines turn out parallel? How can you tell from the equations?

Harder version: draw two cards for the slope and make a fraction m = first card ÷ second card.

What to look for: lines are parallel exactly when the two m cards have the same value and the same colour.

Your turn

State the slope and y-intercept of y = 4x − 7.

State the slope and y-intercept of y = −(1/2)x + 3.

State the slope and y-intercept of y = 6 − 2x.

Write the equation of the line with slope 3 and y-intercept −2.

Describe how to graph y = (3/4)x + 1, and name three points on it.

Is the point (2, 5) on the line y = 3x − 1?

Which two of these lines are parallel? y = 2x + 1, y = 3x + 1, y = 2x − 5

More practice: warm-up

What is the y-intercept of y = 5x + 9?

What are the slope and y-intercept of y = −3x?

Write the equation of the line with m = 1 and b = 0.

At what point does y = 2x + 6 cross the y-axis?

Find the x-intercept of y = 2x + 6.

More practice: on your own

Write 5x + y = 2 in the form y = mx + b, and state m and b.

Write 3y = 9x + 12 in the form y = mx + b.

A rental hall charges C = 75h + 200 dollars, where h is the number of hours. What do 75 and 200 mean? What does a 4-hour party cost?

Find both intercepts of y = −2x + 8.

List four points you could plot to graph y = −(1/3)x + 2.

Challenge

Write 4x − 2y = 12 in the form y = mx + b.

Find the equation of the line parallel to y = −4x + 1 that passes through (2, 3).

A line y = mx + b has b > 0 and m < 0. Which quadrants does it pass through? Explain.

Check your understanding

Quiz

What is the slope of y = −5x + 2?

What is the y-intercept of y = 7 − 3x?

Which equation has slope 2 and y-intercept −4?

To graph y = (2/5)x − 1, you start at (0, −1) and then move:

Which line is parallel to y = 3x − 2?

2y = 6x + 8 written in the form y = mx + b is:

What is the x-intercept of y = 3x − 9?

For parents and teachers

Big idea: The equation y = mx + b packs a whole linear relation into two numbers: the slope m (rate of change) and the y-intercept b (initial value). Expectation C4.4 is about connecting rates of change to slopes and initial values to y-intercepts; C3.2 is about moving between the representations.

Common difficulties:

Reading the slope from y = 6 − x as 6. The slope is the coefficient of x, including its sign.
Dividing only some terms when rearranging.
Graphing a fractional slope upside down (run 2, rise 5 instead of rise 2, run 5).

How to help: The card-deck activity makes practice into a game. Ask your child to check each graph by substituting one point. Solving equations (for the x-intercept) is taught in the separate algebra unit on expressions and equations.