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Solve the system of equations:

[tex]\[ \left\{\begin{array}{l}y=15x+9 \\ y=\frac{1}{2}x-20\end{array}\right. \][/tex]

What is the solution to the system?

Enter your answer as an ordered pair, like this: [tex]\((42, 53)\)[/tex].

If your answer includes one or more fractions, use the / symbol to separate numerators and denominators. For example, if your answer is [tex]\(\left(\frac{42}{53}, \frac{64}{75}\right)\)[/tex], enter it like this: [tex]\((42 / 53, 64 / 75)\)[/tex].

If there is no solution, enter "no"; if there are infinitely many solutions, enter "inf."



Answer :

GinnyAnswer
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]

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