To solve the system of equations given by:
[tex]\[\begin{cases}
y = 15x + 9 \\
y = \frac{1}{2}x - 20
\end{cases}\][/tex]
We start by setting the expressions for [tex]\( y \)[/tex] equal to each other since they both represent [tex]\( y \)[/tex]:
[tex]\[ 15x + 9 = \frac{1}{2}x - 20 \][/tex]
Next, we will eliminate the fraction by multiplying every term by 2 to make calculation easier:
[tex]\[ 2(15x + 9) = 2 \left(\frac{1}{2}x - 20\right) \][/tex]
This simplifies to:
[tex]\[ 30x + 18 = x - 40 \][/tex]
We then move all the [tex]\( x\)[/tex]-terms to one side and the constant terms to the other side:
[tex]\[ 30x - x = -40 - 18 \][/tex]
Simplifying both sides yields:
[tex]\[ 29x = -58 \][/tex]
We solve for [tex]\( x \)[/tex] by dividing both sides by 29:
[tex]\[ x = \frac{-58}{29} \][/tex]
Which simplifies to:
[tex]\[ x = -2 \][/tex]
With this value of [tex]\( x \)[/tex], we substitute back into one of the original equations to find [tex]\( y \)[/tex]. Let’s use the equation [tex]\( y = 15x + 9 \)[/tex]:
[tex]\[ y = 15(-2) + 9 \][/tex]
Which simplifies to:
[tex]\[ y = -30 + 9 \][/tex]
[tex]\[ y = -21 \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
\boxed{(-2, -21)}
\][/tex]
