To solve the system of equations by the substitution method, we must substitute one equation into the other to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Here are the steps:
1. Given Equations:
[tex]\[
(1) \quad x = 5y + 2
\][/tex]
[tex]\[
(2) \quad x = 9y - 10
\][/tex]
2. Equate the expressions for [tex]\( x \)[/tex] from both equations:
[tex]\[
5y + 2 = 9y - 10
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
- Move the terms involving [tex]\( y \)[/tex] to one side of the equation:
[tex]\[
5y + 2 - 9y = -10
\][/tex]
[tex]\[
-4y + 2 = -10
\][/tex]
- Move the constant term to the other side:
[tex]\[
-4y = -10 - 2
\][/tex]
[tex]\[
-4y = -12
\][/tex]
- Divide both sides by [tex]\(-4\)[/tex]:
[tex]\[
y = \frac{-12}{-4}
\][/tex]
[tex]\[
y = 3
\][/tex]
4. Substitute the value of [tex]\( y \)[/tex] back into one of the original equations to find [tex]\( x \)[/tex]:
- Using equation (1):
[tex]\[
x = 5y + 2
\][/tex]
[tex]\[
x = 5(3) + 2
\][/tex]
[tex]\[
x = 15 + 2
\][/tex]
[tex]\[
x = 17
\][/tex]
5. Write the solution set:
[tex]\[
(x, y) = (17, 3)
\][/tex]
Therefore, the solution set is:
[tex]\[ \boxed{(17, 3)} \][/tex]
Select the correct choice:
A. The solution set is [tex]\((17, 3)\)[/tex].
B. There are infinitely many solutions.
C. There is no solution.
