Question

A system of equations is created by using the line that is created by the equation 3 x minus 2 y = negative 4 and the line that is created by the data in the table below.

x
y
–3
–9
–1
–5
3
3
5
7

What is the y-value of the solution to the system?

Answer 1

To find the y-value of the solution to the system of equations formed by the equation \( 3x - 2y = -4 \) and the line represented by the data table, we first need to determine the equation of the line created by the points in the table.

The points given in the table are:
- (-3, -9)
- (-1, -5)
- (3, 3)
- (5, 7)

We can first calculate the slope (\( m \)) of the line using two of the points. Let's use the points (-3, -9) and (5, 7).

The formula for slope \( m \) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points (-3, -9) \((x_1, y_1)\) and (5, 7) \((x_2, y_2)\):
\[
m = \frac{7 - (-9)}{5 - (-3)} = \frac{7 + 9}{5 + 3} = \frac{16}{8} = 2
\]

Now that we have the slope, we will find the equation of the line in point-slope form using the slope and one of the points. Let's use the point (-3, -9).

The point-slope form is:
\[
y - y_1 = m(x - x_1)
\]
Substituting the values:
\[
y - (-9) = 2(x - (-3)) \Rightarrow y + 9 = 2(x + 3)
\]
Expanding this gives:
\[
y + 9 = 2x + 6 \Rightarrow y = 2x + 6 - 9 \Rightarrow y = 2x - 3
\]

Now we have the equation of the line from the table: **\( y = 2x - 3 \)**.

Next, we will set this equal to the rearranged form of the first equation \( 3x - 2y = -4 \) by solving for \( y \):
\[
3x - 2y = -4 \Rightarrow -2y = -3x - 4 \Rightarrow 2y = 3x + 4 \Rightarrow y = \frac{3}{2}x + 2
\]

Now, we have the two equations we need to solve:
1. \( y = 2x - 3 \)
2. \( y = \frac{3}{2}x + 2 \)

Setting the two equations equal to each other:
\[
2x - 3 = \frac{3}{2}x + 2
\]

To eliminate the fraction, multiply every term by 2:
\[
4x - 6 = 3x + 4
\]

Now, solve for \( x \):
\[
4x - 3x = 4 + 6 \Rightarrow x = 10
\]

Now substitute \( x = 10 \) back into one of the equations to find \( y \). Using \( y = 2x - 3 \):
\[
y = 2(10) - 3 = 20 - 3 = 17
\]

Thus, the y-value of the solution to the system is **17**.