Question

identify the choice that best completes the statement or answers the question.
____ 31. write the equation $-x + 4y = -12$ in slope - intercept form. then graph the equation.
a. $y = \frac{1}{4}x - 3$
c. $y = \frac{1}{4}x - 3$
b. $y = \frac{1}{4}x - 3$
d. $y = \frac{1}{4}x - 3$
____ 32. the equation of a linear function that passes through the point $(3, -1)$ and has a slope of -2 can be written in point - slope form as $y + 1 = -2(x - 3)$.
which equation can be used to best identify the y - intercept of the linear function?
a. $y = 2x - 7$
c. $2x + 2y = 7$
b. $y = -2x + 5$
d. $2x + 2y = -5$

Explanation:

Question 31

Step 1: Recall slope - intercept form

The slope - intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept. We need to rewrite the equation \(-x + 4y=-12\) in this form.
First, isolate \(y\). Add \(x\) to both sides of the equation: \(4y=x - 12\).

Step 2: Solve for \(y\)

Divide each term in the equation \(4y=x - 12\) by \(4\): \(y=\frac{1}{4}x-3\).
Now, let's analyze the graph of the line \(y = \frac{1}{4}x-3\). The slope \(m=\frac{1}{4}\) (positive, so the line should rise from left to right) and the \(y\) - intercept \(b=-3\) (the line crosses the \(y\) - axis at \((0, - 3)\)).
Looking at the options:

• Option a: The graph seems to have a wrong slope or intercept (not consistent with \(y=\frac{1}{4}x - 3\)).
• Option b: The graph is not a straight line (it's a curve), so it's not a linear graph.
• Option c: The line has a negative slope (since it's decreasing from left to right), but our slope is positive, so this is incorrect.
• Option d: The line has a positive slope (\(\frac{1}{4}\)) and the \(y\) - intercept at \((0,-3)\), which matches the equation \(y = \frac{1}{4}x-3\).

The point - slope form of the linear function is \(y + 1=-2(x - 3)\). To find the \(y\) - intercept, we can convert the point - slope form to slope - intercept form (\(y=mx + b\), where \(b\) is the \(y\) - intercept) or we can set \(x = 0\) in the equation and solve for \(y\). But first, let's expand the point - slope form.
Expand \(y + 1=-2(x - 3)\):
\(y+1=-2x + 6\)
Subtract 1 from both sides: \(y=-2x+5\)
Now, let's check the options:

• Option a: \(y = 2x-7\) has a slope of \(2\), but our slope is \(-2\), so it's incorrect.
• Option b: \(y=-2x + 5\) is in slope - intercept form. If we want to find the \(y\) - intercept, we can see that when \(x = 0\), \(y = 5\). This equation is obtained by converting the point - slope form to slope - intercept form, which is the best way to identify the \(y\) - intercept.
• Option c: \(2x + 2y=7\) can be rewritten as \(y=-x+\frac{7}{2}\), which has a different slope and intercept, so it's incorrect.
• Option d: \(2x + 2y=-5\) can be rewritten as \(y=-x-\frac{5}{2}\), which has a different slope and intercept, so it's incorrect.

Answer:

d. \(y=\frac{1}{4}x - 3\) (with the corresponding graph)

Question 32