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Use the substitution method to solve the following system of equations.

[tex]\[
\begin{cases}
2x - y = 13 \\
4x + 3y = -9
\end{cases}
\][/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice:

A. There is one solution. The solution of the system is [tex]\(\square\)[/tex].

(Simplify your answer. Type an ordered pair.)

B. The solution set of the system is [tex]\(\{(x, y) \mid 2x - y = 13\}\)[/tex].

C. The solution set is [tex]\(\varnothing\)[/tex].



Answer :

GinnyAnswer
To solve the system of equations using the substitution method, follow these steps.

Given the system of equations:
[tex]\[ \left\{\begin{array}{l} 2x - y = 13 \\ 4x + 3y = -9 \end{array}\right. \][/tex]

Step 1: Solve one of the equations for one of the variables. Here we can start with the first equation. Let's isolate [tex]\( y \)[/tex]:

[tex]\[ 2x - y = 13 \][/tex]
[tex]\[ -y = 13 - 2x \][/tex]
[tex]\[ y = 2x - 13 \][/tex]

Step 2: Substitute this expression for [tex]\( y \)[/tex] into the second equation:

[tex]\[ 4x + 3(2x - 13) = -9 \][/tex]

Step 3: Simplify and solve for [tex]\( x \)[/tex]:

[tex]\[ 4x + 6x - 39 = -9 \][/tex]
[tex]\[ 10x - 39 = -9 \][/tex]
[tex]\[ 10x = -9 + 39 \][/tex]
[tex]\[ 10x = 30 \][/tex]
[tex]\[ x = 3 \][/tex]

Step 4: Substitute [tex]\( x = 3 \)[/tex] back into the equation we used to express [tex]\( y \)[/tex]:

[tex]\[ y = 2(3) - 13 \][/tex]
[tex]\[ y = 6 - 13 \][/tex]
[tex]\[ y = -7 \][/tex]

So, the solution to the system of equations is:
[tex]\[ (3, -7) \][/tex]

Therefore, the correct choice is:

A. There is one solution. The solution of the system is [tex]\((3, -7)\)[/tex].

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