Question

① the table reps. points on graph of linear function; which reps. this function?
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$$\begin{array}{c|c|c|c} x & -4 & -2 & 6 \\\\ hline y & 10 & 7 & -5 \\end{array}$$

a) $y = -\frac{3}{2}x + 4$ c) $y = -\frac{2}{3}x - 9$
b) $y = -\frac{3}{2}x - 14$ d) $y = -\frac{2}{3}x - 1$

② which is the solution to this system of equations?
$-3x + 5y = 21$
$6x - y = -15$
a) $(2, 27)$ c) $(2, 0)$
b) $(-2, 3)$ d) $(-3, -3)$

③ which is the value of $x$ in the solution to this system of equations?
$3x - 5y = 22$
$y = -5x + 32$
a) $-6.5$ c) $6.5$
b) $0.5$ d) $-0.5$

④ which situation shows causation?
a) when the number of people in bus increases, the number of animals in a zoo also increases
b) when the number of hours worked each week by an hourly employee decreases, the amount of money earned by the employee also decreases
c) when the amount of a discount for a sale increases, the number of items sold during sale decreases
d) when the number of bike trails in a city decreases, the amount of rainfall in the city increases

Explanation:

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Problem 1

Step1: Calculate slope $m$

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{7-10}{-2-(-4)}=\frac{-3}{2}=-\frac{3}{2}$

Step2: Find y-intercept $b$

Use $(x,y)=(-4,10)$: $10=-\frac{3}{2}(-4)+b \implies 10=6+b \implies b=4$

Step3: Form linear equation

$y=-\frac{3}{2}x + 4$
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Problem 2

Step1: Isolate $y$ in second equation

$y=6x+15$

Step2: Substitute into first equation

$-3x+5(6x+15)=21 \implies -3x+30x+75=21$

Step3: Solve for $x$

$27x=21-75=-54 \implies x=-2$

Step4: Solve for $y$

$y=6(-2)+15=3$
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Problem 3

Step1: Substitute $y$ into first equation

$3x-5(-5x+32)=22 \implies 3x+25x-160=22$

Step2: Solve for $x$

$28x=22+160=182 \implies x=\frac{182}{28}=6.5$
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Problem 4

Step1: Define causation

Causation = one event directly causes another

Step2: Evaluate options

Only option b has a direct causal link: fewer work hours (hourly) directly causes lower earnings. Other options are correlations or unrelated relationships.

Answer:

1. a) $y=-\frac{3}{2}x + 4$
2. b) $(-2, 3)$
3. c) $6.5$
4. b) When the number of hours worked each week by an hourly employee decreases, the amount of money earned by the employee also decreases