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].
