Math Test Prep

UNIT 3 · LINEAR EQUATIONS

Let’s make lines
make sense.

Learn one idea at a time, try it right away, then check what you know with a full practice test.

Find my starting point Start with slope
6 short lessons9 practice skills16 test questions
A linear equation makes a straight line.

Its slope describes how y changes when x changes. Its intercept tells you where the line starts.

y = mx + b

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LESSON 01THE RATE

Slope: how steep, and which way?

Slope measures the change in y for each change in x. Think “rise over run.”

Slope-intercept form

y = mx + b

m is the slope. b is the y-intercept, the y-value when x = 0.

WORKED EXAMPLE

For y = 2x + 3, the slope is 2 and the y-intercept is (0, 3). From that point, go up 2 and right 1.

From two points

m = (y₂ − y₁) ÷ (x₂ − x₁)

Subtract the y-values in the same order as the x-values. Never divide by zero.

WORKED EXAMPLE

Points (1, 2) and (4, 8): m = (8 − 2) ÷ (4 − 1) = 6 ÷ 3 = 2.

From a table or graph

Pick two rows or two clear points, then use the same rise-over-run formula. A graph’s slope is not how high the line is; it is how much it changes as you move right.

TABLE EXAMPLE

If x goes from 0 to 2 while y goes from 5 to 1, then m = (1 − 5) ÷ (2 − 0) = −2.

Positiverises to the right
Negativefalls to the right
Zerohorizontal line
Undefinedvertical line
LESSON 02THE EQUATION

Rewrite an equation to see its slope

Put y by itself to turn standard form into slope-intercept form. Point-slope form is useful when you know a point and a slope.

Rearrange into y = mx + b

Ax + By = C
  1. Move the x-term to the other side.
  2. Divide every term by the coefficient of y.
  3. Read the slope and y-intercept.
WORKED EXAMPLE

2x + 3y = 12
3y = −2x + 12
y = −⅔x + 4

The slope is −⅔ and the y-intercept is (0, 4).

Point-slope form

y − y₁ = m(x − x₁)

Use the slope m and a known point (x₁, y₁). Keep each coordinate’s sign inside its parentheses.

WORKED EXAMPLE

Slope 3 through (2, 5):
y − 5 = 3(x − 2)

To find slope-intercept form, distribute and solve for y.

LESSON 03THE PICTURE

Graph a line and find its intercepts

Start at the y-intercept. Use slope as rise over run to find another point, then draw the straight line through them.

Plot y = mx + b

  1. Plot (0, b) on the y-axis.
  2. Write slope as a fraction: rise ÷ run.
  3. Move from the first point and plot again.
  4. Draw a straight line through the points.
WORKED EXAMPLE

For y = −½x + 2, plot (0, 2), then go down 1 and right 2 to (2, 1).

Find both intercepts

The y-intercept is where the line crosses the y-axis, so set x = 0. The x-intercept is where it crosses the x-axis, so set y = 0 and solve.

WORKED EXAMPLE

For y = 2x + 6: y-intercept is (0, 6). Set y = 0: 0 = 2x + 6, so x = −3 and the x-intercept is (−3, 0).

TRY THE GRAPH

Change m or b and watch the line move.

The grid shows points from −5 to 5. The equation updates as you change the controls.

y = x
LESSON 04INPUTS & OUTPUTS

Domain and range

The domain is the set of allowed x-values. The range is the set of y-values the relation can produce.

For a line

A non-vertical line keeps going left and right, so its domain is all real numbers. If it is not horizontal, it also reaches every y-value, so its range is all real numbers.

Domain → x-values
Range → y-values

In a real situation

Context can limit the answers. If x counts tickets, negative x-values do not make sense. Check whether the question asks for whole-number inputs or only a certain time period.

WORKED EXAMPLE

A 0–10 hour rental has domain 0 ≤ h ≤ 10. If cost is C = 4h + 8, the range is 8 ≤ C ≤ 48.

LESSON 05COMPARE LINES

Parallel and perpendicular lines

Parallel: same slope

Parallel lines rise or fall at the same rate. They have equal slopes and different intercepts, so they never meet.

WORKED EXAMPLE

A line with slope ⅔ is parallel to any line with slope ⅔.

Perpendicular: negative reciprocals

Flip the fraction and change its sign. The slopes multiply to −1, and the lines meet at a right angle.

WORKED EXAMPLE

If one slope is ¾, the perpendicular slope is −⁴⁄₃.

LESSON 06USE THE MATH

Model a story. Read an inequality.

Linear word problems

Look for a starting amount and a rate that repeats. Let x be the number of steps or items. Let y be the total.

WORKED EXAMPLE

A ride card starts with $5 and adds $2 per ride. Total balance used is y = 2x + 5. The slope is $2 per ride; the starting amount is $5.

Inequality symbols

< less than open / dashed boundary

≤ less than or equal closed / solid boundary

> greater than open / dashed boundary

≥ greater than or equal closed / solid boundary

For y > mx + b, shade above the boundary line. For y < mx + b, shade below it.

YOUR TURN

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Final practice test

16 questions across all six lessons. Work through the full test, then review every answer and explanation.

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