Let's solve the system of equations step by step.
Step 1: Write down the equations
We have the following system of equations:
[tex]\[
\frac{x + y}{7} = \frac{x}{2}
\][/tex]
[tex]\[
\frac{3}{2}x + \frac{2}{5}y = 5
\][/tex]
Step 2: Clear fractions from the first equation
Multiply both sides of the first equation by 14 (the least common multiple of 7 and 2) to eliminate the fractions:
[tex]\[
14 \cdot \frac{x + y}{7} = 14 \cdot \frac{x}{2}
\][/tex]
This simplifies to:
[tex]\[
2(x + y) = 7x
\][/tex]
Distribute:
[tex]\[
2x + 2y = 7x
\][/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[
2y = 7x - 2x
\][/tex]
[tex]\[
2y = 5x
\][/tex]
[tex]\[
y = \frac{5}{2}x
\][/tex]
Step 3: Substitute [tex]\( y \)[/tex] into the second equation
Now substitute [tex]\( y = \frac{5}{2}x \)[/tex] into the second equation:
[tex]\[
\frac{3}{2}x + \frac{2}{5}\left(\frac{5}{2}x\right) = 5
\][/tex]
Simplify the terms:
[tex]\[
\frac{3}{2}x + \frac{2}{5} \cdot \frac{5}{2}x = 5
\][/tex]
[tex]\[
\frac{3}{2}x + x = 5
\][/tex]
Combine the [tex]\( x \)[/tex] terms:
[tex]\[
\frac{5}{2}x = 5
\][/tex]
Multiply both sides by [tex]\( \frac{2}{5} \)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 2
\][/tex]
Step 4: Find [tex]\( y \)[/tex]
Use the value of [tex]\( x \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[
y = \frac{5}{2}x
\][/tex]
[tex]\[
y = \frac{5}{2} \cdot 2
\][/tex]
[tex]\[
y = 5
\][/tex]
So, the solution to the system of equations is:
[tex]\[
(x, y) = (2, 5)
\][/tex]
