Math 10C · Linear functions
Slope, and three ways to write a line
A line is the graph of a constant rate of change. Everything in this unit is either that rate — the slope — or a different way of writing the same line down.
- 1. The words, first
- 2. Slope as a rate
- 3. Three forms, and when to use each
- 4. Systems of equations
- 5. What costs marks
The words, first
The idea: Three forms, and the fact each one gives you for free.
| Word | What it means |
|---|---|
| Slope (m) | Rise over run: the change in y for each unit change in x. In a real situation it is a rate, with units. |
| y-intercept (b) | Where the line crosses the y-axis. The value when x = 0. |
| x-intercept | Where it crosses the x-axis. The value when y = 0. |
| Slope-intercept form | y = mx + b. Hands you the slope and the y-intercept. |
| General form | Ax + By + C = 0, with A a whole number and usually positive. Tidy for writing down; hands you nothing directly. |
| Slope-point form | y − y₁ = m(x − x₁). Built for when you know the slope and any one point. |
| Parallel lines | Equal slopes, different intercepts. They never meet. |
| Perpendicular lines | Slopes are negative reciprocals, so their product is −1. |
| System of equations | Two or more equations solved together. A solution satisfies all of them at once, so it is a point. |
Slope as a rate
The idea: Slope is not just the steepness of a picture. In any real situation it is how fast one quantity changes with another, and it carries units.
m = (y₂ − y₁) / (x₂ − x₁)
Through (1, 2) and (3, 8): m = (8 − 2)/(3 − 1) = 3. Subtract the coordinates in the same order top and bottom — swapping one flips the sign.
Special cases. A horizontal line has no rise, so its slope is 0. A vertical line has no run, so its slope is undefined — division by zero, not zero.
As a rate. If a graph shows cost against hours, the slope is dollars per hour and the intercept is the call-out fee. Being able to say what the slope means in the situation is worth more marks than calculating it.
Parallel and perpendicular. Parallel means equal slopes. Perpendicular means flip the fraction and change the sign: the perpendicular to 3/4 has slope −4/3, and 3/4 × −4/3 = −1 ✓.
Three forms, and when to use each
The idea: They describe the same line. Choose by what you were given, not by habit.
| Form | Use it when | It gives you |
|---|---|---|
| y = mx + b | You know the slope and the y-intercept | Slope and intercept, and it graphs instantly |
| y − y₁ = m(x − x₁) | You know the slope and any point | A direct write-down with no algebra |
| Ax + By + C = 0 | The question asks for it | No fractions; slope is −A/B |
Worked. The line through (2, 1) and (6, 9). Slope = (9 − 1)/(6 − 2) = 2. Slope-point form gives y − 1 = 2(x − 2) straight away, which rearranges to y = 2x − 3. Checking the other point is free: 2(6) − 3 = 9 ✓.
From general form. 2x + 3y = 12 rearranges to y = −(2/3)x + 4, so the slope is −2/3. Or read it straight off as −A/B.
Systems of equations
The idea: Two lines, one question: where do they agree? The answer is a point, so it has two coordinates.
By substitution. y = x + 4 and 2x + y = 10. Replace y: 2x + (x + 4) = 10, so 3x = 6, x = 2 and y = 6. The solution is (2, 6).
By elimination. 3x + 2y = 16 and 3x − y = 1. Both start with 3x, so subtracting removes it: 3y = 15, y = 5, then x = 2. The solution is (2, 5).
Always check in the other equation. 3(2) + 2(5) = 16 ✓. A solution that satisfies only one equation is not a solution.
How many solutions are possible?
- One — different slopes, so the lines cross once.
- None — same slope, different intercepts: parallel, never meeting.
- Infinitely many — same slope and same intercept: the same line written twice.
Comparing the slopes before solving tells you which case you are in, and sometimes answers the question outright.
What costs marks
The idea: Signs and half-answers.
- Subtracting coordinates in different orders top and bottom, which flips the slope's sign.
- Calling a vertical line's slope zero. It is undefined.
- Giving only the x-coordinate as the solution to a system.
- Not checking in both equations.