Straight-line graphs: solved examples

Worked examples of straight-line graphs, from easy to hard, with every step written out. Try each one yourself first, then check the working.

✓ Every answer here is computed and checked by the math engine

Easy

Example 1. What is the y-intercept of the line $y = x - 5$ ?
1. Match the equation to y = mx + c. $m = 1, \quad c = -5$
Answer: $-5$
Example 2. What is the gradient of the line $y = -9x - 5$ ?
1. Match the equation to y = mx + c. $m = -9, \quad c = -5$
Answer: $-9$
Example 3. What is the gradient of the line $y = 5x + 9$ ?
1. Match the equation to y = mx + c. $m = 5, \quad c = 9$
Answer: $5$

Example 4. What is the gradient of the line through $(1, 8)$ and $(-3, 0)$ ?
1. Change in y over change in x. $m = \frac{0 - 8}{-3 - 1} = \frac{-8}{-4}$
2. Simplify. $m = 2$
Answer: $2$
Example 5. What is the gradient of the line through $(-6, -7)$ and $(-3, 6)$ ?
1. Change in y over change in x. $m = \frac{6 - (-7)}{-3 - (-6)} = \frac{13}{3}$
2. Simplify. $m = \frac{13}{3}$
Answer: $\frac{13}{3}$
Example 6. What is the gradient of the line through $(4, -9)$ and $(5, 9)$ ?
1. Change in y over change in x. $m = \frac{9 - (-9)}{5 - 4} = \frac{18}{1}$
2. Simplify. $m = 18$
Answer: $18$

Example 7. A line has gradient $2$ and passes through $(1, 8)$ . What is its y-intercept?
1. Put the point into y = mx + c. $8 = 2 \times 1 + c$
2. Solve for c. $c = 8 - 2 = 6$
Answer: $6$
Example 8. A line has gradient $-2$ and passes through $(-6, -7)$ . What is its y-intercept?
1. Put the point into y = mx + c. $-7 = -2 \times (-6) + c$
2. Solve for c. $c = -7 - 12 = -19$
Answer: $-19$
Example 9. A line has gradient $-1$ and passes through $(4, -9)$ . What is its y-intercept?
1. Put the point into y = mx + c. $-9 = -1 \times 4 + c$
2. Solve for c. $c = -9 - (-4) = -5$
Answer: $-5$

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