Question

Name another point on line q, which is parallel to the line y-3=\frac{5}{2}\left(x+1\right) and goes through the point (2, -4)

Answer 1

To find a point on line \( q \), which is parallel to the line given by the equation \( y - 3 = \frac{5}{2}(x + 1) \) and passes through the point \( (2, -4) \), we first need to identify the slope of the given line.

The equation \( y - 3 = \frac{5}{2}(x + 1) \) can be rewritten in slope-intercept form \( y = mx + b \):

1. Expanding the equation:
\[
y - 3 = \frac{5}{2}x + \frac{5}{2}
\]

2. Adding 3 to both sides:
\[
y = \frac{5}{2}x + \frac{5}{2} + 3
\]
\[
y = \frac{5}{2}x + \frac{5}{2} + \frac{6}{2}
\]
\[
y = \frac{5}{2}x + \frac{11}{2}
\]

From the slope-intercept form, we see that the slope \( m \) of the original line is \( \frac{5}{2} \).

Since line \( q \) is parallel to the given line, it will have the same slope \( \frac{5}{2} \).

Now, we can use the point-slope form to find the equation of line \( q \), which goes through the point \( (2, -4) \):

The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
Substituting \( x_1 = 2 \), \( y_1 = -4 \), and \( m = \frac{5}{2} \):
\[
y - (-4) = \frac{5}{2}(x - 2)
\]
\[
y + 4 = \frac{5}{2}(x - 2)
\]

Now, we can pick another value for \( x \) to find a corresponding \( y \). Let's choose \( x = 0 \):

Substituting \( x = 0 \){
\[
y + 4 = \frac{5}{2}(0 - 2)
\]
\[
y + 4 = \frac{5}{2}(-2)
\]
\[
y + 4 = -5
\]
\[
y = -5 - 4
\]
\[
y = -9
\]

Thus, the point on line \( q \), which is parallel to the original line and passes through \( (2, -4) \), is \( (0, -9) \).

So, another point on line \( q \) is \( \boxed{(0, -9)} \).