To solve the system of linear equations using the substitution method, follow these steps:
Given the system:
[tex]\[
\begin{array}{l}
5x - 2y = 49 \quad \text{(Equation 1)} \\
-6x + y = -49 \quad \text{(Equation 2)}
\end{array}
\][/tex]
1. Solve Equation 2 for [tex]\(y\)[/tex]:
[tex]\[
-6x + y = -49
\][/tex]
[tex]\[
y = 6x - 49
\][/tex]
2. Substitute [tex]\(y\)[/tex] in Equation 1:
Replace [tex]\(y\)[/tex] in Equation 1 with the expression obtained from Equation 2:
[tex]\[
5x - 2(6x - 49) = 49
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
5x - 12x + 98 = 49
\][/tex]
Combine like terms:
[tex]\[
-7x + 98 = 49
\][/tex]
Isolate [tex]\(x\)[/tex] by first subtracting 98 from both sides:
[tex]\[
-7x = 49 - 98
\][/tex]
[tex]\[
-7x = -49
\][/tex]
Divide both sides by -7:
[tex]\[
x = 7
\][/tex]
3. Substitute [tex]\(x = 7\)[/tex] back into the expression for [tex]\(y\)[/tex]:
Use the value of [tex]\(x\)[/tex] in the equation [tex]\( y = 6x - 49 \)[/tex]:
[tex]\[
y = 6(7) - 49
\][/tex]
Simplify the right-hand side:
[tex]\[
y = 42 - 49
\][/tex]
[tex]\[
y = -7
\][/tex]
So, the solution to the system of equations is:
[tex]\[
(x, y) = (7, -7)
\][/tex]
Therefore, the system has one solution:
[tex]\[
\boxed{(7, -7)}
\][/tex]
