To solve the system of linear equations
[tex]\[
3x + y = 2 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
6x - y = 25 \quad \text{(Equation 2)}
\][/tex]
we can use the method of elimination or substitution. Here's a detailed step-by-step solution using the elimination method:
1. Add the two equations to eliminate [tex]\( y \)[/tex]:
[tex]\[
(3x + y) + (6x - y) = 2 + 25
\][/tex]
2. This simplifies to:
[tex]\[
3x + y + 6x - y = 27
\][/tex]
3. Combine like terms:
[tex]\[
9x = 27
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{27}{9}
\][/tex]
[tex]\[
x = 3
\][/tex]
5. Now that we have [tex]\( x \)[/tex], substitute it back into one of the original equations to solve for [tex]\( y \)[/tex]. We can choose Equation 1:
[tex]\[
3(3) + y = 2
\][/tex]
[tex]\[
9 + y = 2
\][/tex]
6. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = 2 - 9
\][/tex]
[tex]\[
y = -7
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
x = 3
\][/tex]
[tex]\[
y = -7
\][/tex]
So, the solution is [tex]\( (3, -7) \)[/tex].
