Question

question 2.
9x - 12y = 36
select all correct statements about the key features of the linear equation.
a. the y-intercept is (0, - 12) .
b. the maximum value of the function is 36.
c. the point (-8, -9) lies on the graph of the function.
d. the x-intercept is (4, 0) .
e. the slope is -\frac{3}{4}.

Explanation:

Step1: Rewrite the equation in slope - intercept form ($y = mx + b$)

Start with the equation $9x-12y = 36$.
Subtract $9x$ from both sides: $- 12y=-9x + 36$.
Divide each term by $-12$: $y=\frac{-9}{-12}x+\frac{36}{-12}$, which simplifies to $y=\frac{3}{4}x - 3$.

Step2: Analyze option A (y - intercept)

The y - intercept of a line in the form $y=mx + b$ is $(0,b)$. From $y=\frac{3}{4}x - 3$, the y - intercept is $(0,-3)$, not $(0, - 12)$. So option A is incorrect.

Step3: Analyze option B (maximum value)

The equation $y=\frac{3}{4}x - 3$ is a linear function with a slope $\frac{3}{4}
eq0$. Linear functions (except constant functions) do not have a maximum or minimum value (they are either increasing or decreasing over the entire real line). So option B is incorrect.

Step4: Analyze option C (check if $(-8,-9)$ lies on the line)

Substitute $x=-8$ and $y = - 9$ into the original equation $9x-12y$.
Left - hand side: $9\times(-8)-12\times(-9)=-72 + 108 = 36$.
The right - hand side of the original equation is 36. Since $9\times(-8)-12\times(-9)=36$, the point $(-8,-9)$ lies on the graph of the function. So option C is correct.

Step5: Analyze option D (x - intercept)

The x - intercept is found by setting $y = 0$ in the original equation $9x-12y=36$.
Substitute $y = 0$: $9x-12\times0=36$, so $9x=36$, and $x = 4$. The x - intercept is $(4,0)$. So option D is correct.

Step6: Analyze option E (slope)

From the slope - intercept form $y=\frac{3}{4}x - 3$, the slope $m=\frac{3}{4}$, not $-\frac{3}{4}$. So option E is incorrect.

Answer:

C. The point $(-8,-9)$ lies on the graph of the function.
D. The x - intercept is $(4,0)$