Question

4.5.6 stepping stones
which of the following statements about the equation $3x + 2y = 10$ are true? select all that apply.

• the slope-intercept form of the equation is $y = 3x + 5$.
• the slope-intercept form of the equation is $y = -dfrac{3}{2}x + 5$.
• the slope is $3$.
• the slope is $-dfrac{3}{2}$.
• the $y$-intercept is $10$.
• the $y$-intercept is $5$.

Explanation:

Step1: 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 \(3x + 2y=10\) in slope - intercept form.
First, isolate \(y\):
Subtract \(3x\) from both sides of the equation \(3x + 2y = 10\):
\(2y=-3x + 10\)
Then, divide each term by \(2\):
\(y=\frac{-3x + 10}{2}=\frac{-3x}{2}+\frac{10}{2}=-\frac{3}{2}x + 5\)

Step2: Analyze each statement

• For the statement "The slope - intercept form of the equation is \(y = 3x+5\)": From Step 1, we know the correct slope - intercept form is \(y=-\frac{3}{2}x + 5\), so this statement is false.
• For the statement "The slope - intercept form of the equation is \(y=-\frac{3}{2}x + 5\)": From Step 1, this statement is true.
• For the statement "The slope is 3": From the equation \(y =-\frac{3}{2}x+5\), the slope \(m =-\frac{3}{2}\), so this statement is false.
• For the statement "The slope is \(-\frac{3}{2}\)": From the equation \(y=-\frac{3}{2}x + 5\), the slope \(m =-\frac{3}{2}\), so this statement is true.
• For the statement "The \(y\) - intercept is 10": From the equation \(y=-\frac{3}{2}x + 5\), the \(y\) - intercept \(b = 5\), so this statement is false.
• For the statement "The \(y\) - intercept is 5": From the equation \(y=-\frac{3}{2}x + 5\), the \(y\) - intercept \(b = 5\), so this statement is true.

Answer:

The slope - intercept form of the equation is \(y =-\frac{3}{2}x + 5\), The slope is \(-\frac{3}{2}\), The \(y\) - intercept is 5. (Corresponding to the options: The slope - intercept form of the equation is \(y =-\frac{3}{2}x + 5\), The slope is \(-\frac{3}{2}\), The \(y\) - intercept is 5)