Graphing Linear Equations: practice solutions, Grade 8

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Practice solutions Grade 8 : Graphing Linear Equations — Zyro the alien explorer of Planète Maths

Written solutions to the chapter problems. Check each step, then correct yourself.

1 Read the slope and the y-intercept ★★★

Compare each equation with $y = mx + b$.

$m = 3$ and $b = 5$.
$m = -2$ and $b = 7$.
$m = \dfrac{1}{2}$ and $b = -4$ (the minus sign belongs to the 4).
$y = -x$ is $y = -1x + 0$, so $m = -1$ and $b = 0$: the line passes through the origin.

2 Complete a table of values ★★★

Substitute each value of $x$.

$x = -1$: $2(-1) - 3 = -5$. $x = 0$: $-3$. $x = 1$: $-1$. $x = 2$: $1$. $x = 3$: $3$.

$x$	$-1$	$0$	$1$	$2$	$3$
$y$	$-5$	$-3$	$-1$	$1$	$3$

Each time $x$ increases by 1, $y$ increases by 2, which is the slope.

3 Is the point on the line? ★★★

Replace $x$ and $y$ in the equation and see if it is true.

$(3, 5)$: $3 + 2 = 5$, true, so the point is on the line.

$(-2, 0)$: $-2 + 2 = 0$, true, so the point is on the line.

$(1, 4)$: $1 + 2 = 3 \neq 4$, so the point is not on the line.

4 Slope from two points ★★★

$m = \dfrac{8 - 2}{4 - 1} = \dfrac{6}{3} = 2$.
$m = \dfrac{2 - 5}{3 - 0} = \dfrac{-3}{3} = -1$.

The first line rises, the second falls.

5 True or false? ★★★

True: all points have the same y-coordinate 4, so the slope is 0.
False: all points have x-coordinate 4, so the line is vertical.
True: the slope $-3$ is negative.
False: $|5| > |0.5|$, so $y = 5x + 1$ is steeper.

6 Build an equation ★★★

$y = -3x + 2$.
$y = 4x - 6$.
$y = 0x + 7$, which is simply $y = 7$: a horizontal line.

7 Saving money ★★★

The starting value is 15 and the rate is 5 per week: $y = 5x + 15$.
$y = 5(6) + 15 = 45$. She has $45.
The slope 5 is the amount saved per week; the y-intercept 15 is the amount she had at the start (week 0).

8 Graph a line from its equation ★★★

$x = -1$: $5$. $x = 0$: $3$. $x = 1$: $1$. $x = 2$: $-1$. $x = 3$: $-3$.

$x$ $-1$ $0$ $1$ $2$ $3$

$y$ $5$ $3$ $1$ $-1$ $-3$
Plot $(0, 3)$. The slope is $-2 = \dfrac{-2}{1}$: go down 2 and right 1 to reach $(1, 1)$, then again to reach $(2, -1)$. Draw the line through these points.

\(x\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)
\(y\)	\(5\)	\(3\)	\(1\)	\(-1\)	\(-3\)

9 Write the equation from the graph ★★★

The line crosses the y-axis at $A(0, -2)$, so $b = -2$.
$m = \dfrac{2 - (-2)}{2 - 0} = \dfrac{4}{2} = 2$.
$y = 2x - 2$.
For $C(1, 0)$: $2(1) - 2 = 0$. The point satisfies the equation.

10 From a table to an equation ★★★

$y$ grows by 3 when $x$ grows by 1, so $m = 3$. At $x = 0$, $y = 4$, so $b = 4$.
$y = 3x + 4$.
$y = 3(10) + 4 = 34$.

11 Linear or not? ★★★

Table 1: $x$ increases by 1 each time, but $y$ increases by 3, then 6, then 12. The change is not constant, so the table is not linear.

Table 2: $x$ increases by 2 and $y$ decreases by 4 each time, so $m = \dfrac{-4}{2} = -2$, constant. The table is linear. At $x = 0$, $y = 10$, so $y = -2x + 10$. Check at $x = 6$: $-12 + 10 = -2$.

12 Horizontal or vertical? ★★★

Every point of the line has y-coordinate $-3$: $y = -3$.
Every point has x-coordinate 4: $x = 4$.
The horizontal line has slope $0$; the slope of the vertical line is undefined.

13 Taxi fare ★★★

$C = 2.25m + 3.5$.
$C = 2.25(8) + 3.5 = 18 + 3.5 = 21.5$. The ride costs $21.50.
$2.25m + 3.5 = 30.5$, so $2.25m = 27$ and $m = 12$. The ride covered 12 miles (about 19.3 km).

14 A fractional slope ★★★

$y = -\dfrac{1}{2}x + 5$.
$y = -\dfrac{1}{2}(6) + 5 = -3 + 5 = 2$.
Plot $(0, 5)$. The slope means down 1 and right 2, giving $(2, 4)$ and then $(4, 3)$.

15 Two points, no intercept ★★★

Slope: $m = \dfrac{16 - 7}{5 - 2} = \dfrac{9}{3} = 3$.

Substitute $(2, 7)$ in $y = 3x + b$: $7 = 6 + b$, so $b = 1$.

The equation is $y = 3x + 1$. Check with $(5, 16)$: $3(5) + 1 = 16$.

16 From standard form to slope-intercept form ★★★

Subtract $2x$ from both sides: $y = -2x + 6$. The slope is $-2$, and $b = 6$.
Subtract $3x$: $4y = -3x + 12$. Divide everything by 4: $y = -\dfrac{3}{4}x + 3$. The slope is $-\dfrac{3}{4}$, and $b = 3$. The line also crosses the x-axis at $(4, 0)$, since $3(4) = 12$.

17 Parallel lines and common points ★★★

Parallel lines have the same slope and different intercepts: $y = 3x + 2$ and $y = 3x - 5$ both have slope 3.
$y = 3x + 2$ and $y = -3x + 2$ share the y-intercept 2, so they meet at $(0, 2)$.
Same slope 3 and $b = -4$: $y = 3x - 4$.

18 Draining a tank ★★★

$g = -15t + 240$.
$g = -15(10) + 240 = 90$ gallons.
$-15t + 240 = 0$, so $t = 16$. The tank is empty after 16 minutes.
The slope $-15$ means the tank loses 15 gallons per minute. 240 gallons is about $240 \times 3.785 \approx 908$ liters.

19 Find the mistake ★★★

Leo used $+2$ instead of $-2$ for the y-intercept. The line starts at $(0, -2)$; going up 3 and right 1 gives $(1, 1)$, then $(2, 4)$.
Rewrite the equation as $y = -2x + 5$. The slope is $-2$ (the coefficient of $x$), and 5 is the y-intercept.

20 Find the missing number ★★★

$13 = 3k + 4$, so $3k = 9$ and $k = 3$.
$-1 = 3(2) + b = 6 + b$, so $b = -7$. The line is $y = 3x - 7$.

21 Choosing a gym ★★★

Gym A: $y = 20x + 10$. Gym B: $y = 25x$.
$20x + 10 = 25x$ gives $10 = 5x$, so $x = 2$. After 2 months both cost $50.
For $x = 6$: Gym A costs $20(6) + 10 = 130$ and Gym B costs $25(6) = 150$. Gym A is cheaper by $20.