Question

Line k has an equation of y=
10
3
x+9. Line 𝓁 includes the point (
–
3,
–
2) and is parallel to line k. What is the equation of line 𝓁?
Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Answer 1

To find the equation of line \( \ell \), which is parallel to line \( k \) and passes through the point \( (-3, -2) \), we first need to identify the slope of line \( k \).

The equation of line \( k \) is given as:

\[
y = \frac{10}{3} x + 9
\]

From this equation, we can see that the slope \( m \) of line \( k \) is \( \frac{10}{3} \).

Since line \( \ell \) is parallel to line \( k \), it will have the same slope. Therefore, the slope of line \( \ell \) is also \( \frac{10}{3} \).

Now, we can use the point-slope form of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]

where \( (x_1, y_1) \) is the point on the line. In this case, the point is \( (-3, -2) \) and the slope \( m \) is \( \frac{10}{3} \).

Substituting in the values, we get:

\[
y - (-2) = \frac{10}{3}(x - (-3))
\]

This simplifies to:

\[
y + 2 = \frac{10}{3}(x + 3)
\]

Next, we distribute \( \frac{10}{3} \):

\[
y + 2 = \frac{10}{3}x + \frac{10}{3} \cdot 3
\]

Since \( \frac{10}{3} \cdot 3 = 10 \), we have:

\[
y + 2 = \frac{10}{3}x + 10
\]

Now, we can isolate \( y \) by subtracting 2 from both sides:

\[
y = \frac{10}{3}x + 10 - 2
\]

This simplifies to:

\[
y = \frac{10}{3}x + 8
\]

Thus, the equation of line \( \ell \) in slope-intercept form is:

\[
\boxed{y = \frac{10}{3}x + 8}
\]