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Find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] if [tex]\((2x + y, x - 2y) = (3, 4)\)[/tex].

Sagot :

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]