Question

the equation of line s is $y = -\frac{1}{2}x + 5$. line t includes the point $(-10, -4)$ and is parallel to line s. what is the equation of line t?
write the equation in slope - intercept form. write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Explanation:

Step1: Determine the slope of line t

Parallel lines have the same slope. The equation of line s is \( y = -\frac{1}{2}x + 5 \), which is in slope - intercept form \( y=mx + b \) (where \( m \) is the slope and \( b \) is the y - intercept). So the slope of line s, \( m_s=-\frac{1}{2} \). Since line t is parallel to line s, the slope of line t, \( m_t = m_s=-\frac{1}{2} \).

Step2: Use the point - slope form to find the equation of line t

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1) \) is a point on the line and \( m \) is the slope of the line. We know that line t passes through the point \( (- 10,-4) \), so \( x_1=-10 \), \( y_1 = - 4 \) and \( m = -\frac{1}{2} \).

Substitute these values into the point - slope form:
\( y-(-4)=-\frac{1}{2}(x - (-10)) \)
Simplify the left - hand side and the right - hand side:
\( y + 4=-\frac{1}{2}(x + 10) \)

Step3: Convert to slope - intercept form

Distribute the \( -\frac{1}{2} \) on the right - hand side:
\( y+4=-\frac{1}{2}x-5 \)
Subtract 4 from both sides of the equation to solve for \( y \):
\( y=-\frac{1}{2}x-5 - 4 \)
\( y=-\frac{1}{2}x-9 \)

Answer:

\( y = -\frac{1}{2}x-9 \)