Sure, let's solve the system of equations step-by-step.
We start with the given system of equations:
1. [tex]\(\frac{x-1}{2} = y - 1\)[/tex]
2. [tex]\(2x = \frac{y}{2} + 5\)[/tex]
Step 1: Simplify the first equation
[tex]\(\frac{x-1}{2} = y - 1\)[/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[ x - 1 = 2(y - 1) \][/tex]
[tex]\[ x - 1 = 2y - 2 \][/tex]
Rearrange to solve for [tex]\(x\)[/tex]:
[tex]\[ x = 2y - 1 + 1 \][/tex]
[tex]\[ x = 2y - 1 + 2 \][/tex]
[tex]\[ x = 2y - 1 \][/tex]
Step 2: Simplify the second equation
[tex]\(2x = \frac{y}{2} + 5\)[/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[ 4x = y + 10 \][/tex]
Rearrange to solve for [tex]\(y\)[/tex]:
[tex]\[ y = 4x - 10 \][/tex]
Step 3: Substitute [tex]\(y = 4x - 10\)[/tex] into the first equation
We already have [tex]\( x = 2y - 1\)[/tex]:
[tex]\[ x = 2(4x - 10) - 1 \][/tex]
[tex]\[ x = 8x - 20 - 1 \][/tex]
[tex]\[ x = 8x - 21 \][/tex]
Now, solve for [tex]\(x\)[/tex]:
[tex]\[ x - 8x = -21 \][/tex]
[tex]\[ -7x = -21 \][/tex]
[tex]\[ x = 3 \][/tex]
Step 4: Substitute [tex]\(x = 3\)[/tex] back into [tex]\(y = 4x - 10\)[/tex]
[tex]\[ y = 4(3) - 10 \][/tex]
[tex]\[ y = 12 - 10 \][/tex]
[tex]\[ y = 2 \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ (x, y) = (3, 2) \][/tex]
