To find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] given the pair of equations [tex]\( (2x + y, x - 2y) = (3, 4) \)[/tex], follow these steps:
1. Set up the system of equations based on the given pairs:
[tex]\[
2x + y = 3
\][/tex]
[tex]\[
x - 2y = 4
\][/tex]
2. Solve the system of equations simultaneously:
First, take the two equations:
[tex]\[
\text{Equation 1: } 2x + y = 3
\][/tex]
[tex]\[
\text{Equation 2: } x - 2y = 4
\][/tex]
Step 1: Solve Equation 2 for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
x = 4 + 2y
\][/tex]
Step 2: Substitute [tex]\( x \)[/tex] from Equation 2 into Equation 1:
[tex]\[
2(4 + 2y) + y = 3
\][/tex]
Simplify the equation:
[tex]\[
8 + 4y + y = 3
\][/tex]
Combine like terms:
[tex]\[
8 + 5y = 3
\][/tex]
Step 3: Solve for [tex]\( y \)[/tex]:
[tex]\[
5y = 3 - 8
\][/tex]
[tex]\[
5y = -5
\][/tex]
[tex]\[
y = -1
\][/tex]
Step 4: Substitute [tex]\( y = -1 \)[/tex] back into Equation 2 to find [tex]\( x \)[/tex]:
[tex]\[
x = 4 + 2(-1)
\][/tex]
Simplify:
[tex]\[
x = 4 - 2
\][/tex]
[tex]\[
x = 2
\][/tex]
Therefore, the values are:
[tex]\[
x = 2 \quad \text{and} \quad y = -1
\][/tex]
