Grade 8 · Grade 8: Linear Equations, Systems and Functions · Linear Relationships and Functions
Linear Equations, Slope and y = mx + c
Solve linear equations with one, no or infinitely many solutions, and connect slope to similar triangles and graphs.
Learning objectives
- Solve linear equations in one variable, including expansion and collecting like terms.
- Recognise equations with no solution or infinitely many solutions.
- Interpret slope as a rate of change.
- Use similar triangles to explain constant slope and derive y = mx + c.
AERO Mathematics alignment
AERO.M8.AE.1
Solve linear equations in one variable, including equations with rational coefficients, expressions requiring expansion and collection of like terms, and equations with one solution, infinitely many solutions or no solution.
AERO.M8.AE.2
Graph proportional relationships, interpret the unit rate as the slope of the graph, and compare two different proportional relationships represented in different ways.
AERO.M8.AE.3
Use similar triangles to explain why the slope is the same between any two distinct points on a non-vertical line, and derive the equations y = mx and y = mx + c.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
Two ramps rise 1 m over a run of 8 m and 1 m over a run of 12 m.
Think about it
Which ramp is steeper, and how can a single number describe steepness?
Hint: Compare rise divided by run.
EXPLAIN
Slope and linear equations
Solving linear equations
Expand brackets, collect like terms, then undo operations. Three outcomes are possible: one solution (3x = 12), no solution (2x + 1 = 2x + 5), or infinitely many solutions (2x + 4 = 2(x + 2)).
Slope as a rate of change
Slope m = rise ÷ run = (y₂ − y₁)/(x₂ − x₁). The first ramp has slope 1/8 = 0.125 and the second 1/12 ≈ 0.083, so the first is steeper.
| Slope | Line looks like | Meaning |
|---|---|---|
| Positive | Rising left to right | Quantity increasing |
| Negative | Falling left to right | Quantity decreasing |
| Zero | Horizontal | No change |
| Undefined | Vertical | Run is zero |
Why slope is constant
Any two points on a line form a right triangle with the axes. All such triangles on the same line are similar, so their rise-to-run ratios are equal. That is exactly why one number describes the whole line, and it leads to y = mx + c, where c is the y-intercept.
Common misconception
No solution and a solution of zero are different. x = 0 is a perfectly good answer; no solution means no value of x can ever work.
INVESTIGATION
Try it yourself
Measure the rise and run of a real ramp, staircase or road slope. Calculate the gradient, write the line as y = mx + c and compare your value with an accessibility guideline such as 1:12.
Watch
Slope from Two Ordered Pairs
Khan Academy · 7:12
Turns rise over run into a reliable calculation.
Before you watch: What is the slope of a horizontal line?
- Find the slope through (2, 5) and (6, 13).
- What does c represent in y = mx + c?
Interactive simulation · PhET
Graphing Slope-Intercept
Change m and c one at a time and record the effect on the line.
While you explore
- What does changing c do to the line?
- Which line is steeper: m = 3 or m = −4? Explain using rise over run.
Key vocabulary
- y-intercept
- Where the line crosses the vertical axis.
- Slope
- How steep a line is.
Practice questions
0/1 correct
Level 1 · Criterion A
Find the slope of the line through (2, 5) and (6, 13).
MYP criterion tasks
Level 2 · Criterion C
Solve 2(x + 3) = 2x + 5 and explain what your result means.
Reflect & track
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