Questions: Line ℓ has the equation y = -(1/2) x + 9. Write the equation of the image of ℓ after a dilation with a scale factor of (1/3), centered at the origin. Write your answer in slope-intercept form.

Line ℓ has the equation y = -(1/2) x + 9. Write the equation of the image of ℓ after a dilation with a scale factor of (1/3), centered at the origin.

Write your answer in slope-intercept form.

Transcript text: Line $\ell$ has the equation $y=-\frac{1}{2} x+9$. Write the equation of the image of $\ell$ after a dilation with a scale factor of $\frac{1}{3}$, centered at the origin. Write your answer in slope-intercept form.

Solution

Solution Steps

Step 1: Identify the Original Line Equation

The original line $ \ell $ is given by the equation: \[ y = -\frac{1}{2} x + 9 \]

Step 2: Apply Dilation

We need to find the image of the line after a dilation with a scale factor of $ \frac{1}{3} $ centered at the origin. For a dilation, we replace $ x $ with $ \frac{1}{\frac{1}{3}} x = 3x $ and $ y $ with $ \frac{1}{\frac{1}{3}} y = 3y $.

Step 3: Substitute and Simplify

Substituting the dilated variables into the original line equation, we have: \[ 3y = -\frac{1}{2}(3x) + 9 \] This simplifies to: \[ 3y = -\frac{3}{2}x + 9 \] Dividing the entire equation by 3 to solve for $ y $: \[ y = 3 - \frac{1}{2}x \]