To solve the given pair of equations and find the point of intersection, let's follow these detailed steps:
1. Identify the equations:
[tex]\[
y_1 = -\frac{3}{2}x - 2
\][/tex]
and we assume the second equation as
[tex]\[
y_2 = 2x + 10
\][/tex]
2. Set the equations equal to each other to find [tex]\(x\)[/tex]:
[tex]\[
-\frac{3}{2}x - 2 = 2x + 10
\][/tex]
3. Collect all [tex]\(x\)[/tex]-terms on one side of the equation:
[tex]\[
-\frac{3}{2}x - 2x = 10 + 2
\][/tex]
4. Combine the [tex]\(x\)[/tex]-terms:
[tex]\[
-\frac{3}{2}x - 2x = -\frac{3}{2}x - \frac{4}{2}x = -\frac{7}{2}x
\][/tex]
So the equation becomes:
[tex]\[
-\frac{7}{2}x = 12
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Multiply both sides by the reciprocal of [tex]\(-\frac{7}{2}\)[/tex]:
[tex]\[
x = 12 \times -\frac{2}{7} = -\frac{24}{7}
\][/tex]
6. Substitute [tex]\(x\)[/tex] back into one of the original equations to find [tex]\(y\)[/tex]. Let's use [tex]\(y_1 = -\frac{3}{2}x - 2\)[/tex]:
[tex]\[
y = -\frac{3}{2}\left(-\frac{24}{7}\right) - 2
\][/tex]
Calculate the multiplication first:
[tex]\[
y = \frac{36}{14} - 2 = \frac{18}{7} - \frac{14}{7} = \frac{4}{7}
\][/tex]
Therefore, the point of intersection of the two equations, which represents the solution to the pair of equations, is:
[tex]\[
\left( -\frac{24}{7}, \frac{4}{7} \right)
\][/tex]
This point in decimal form is approximately:
[tex]\[
(-3.4285714285714284, 0.5714285714285714)
\][/tex]
Plot this point on the coordinate plane to represent the solution of the pair of equations.
