Problem Analysis:
The task involves solving a system of linear equations using the
substitution method. The given system is:
\[
\begin{cases}
x + y = 10 \\
2x - y = 5
\end{cases}
\]
We are asked to solve for \( x \) and \( y \) step by step.
---
Step-by-Step Solution:
####
Step 1: Solve one equation for one variable.
We start by solving the first equation for \( y \):
\[
x + y = 10
\]
Subtract \( x \) from both sides:
\[
y = 10 - x
\]
So, we have expressed \( y \) in terms of \( x \):
\[
y = 10 - x
\]
---
####
Step 2: Substitute the expression for \( y \) into the second equation.
Now, substitute \( y = 10 - x \) into the second equation:
\[
2x - y = 5
\]
Replace \( y \) with \( 10 - x \):
\[
2x - (10 - x) = 5
\]
Simplify the equation:
\[
2x - 10 + x = 5
\]
Combine like terms:
\[
3x - 10 = 5
\]
Add 10 to both sides:
\[
3x = 15
\]
Divide by 3:
\[
x = 5
\]
---
####
Step 3: Solve for \( y \) using the value of \( x \).
Now that we have \( x = 5 \), substitute this value back into the expression for \( y \):
\[
y = 10 - x
\]
Substitute \( x = 5 \):
\[
y = 10 - 5
\]
\[
y = 5
\]
---
####
Step 4: Verify the solution.
To ensure the solution is correct, substitute \( x = 5 \) and \( y = 5 \) back into both original equations:
1. Check the first equation \( x + y = 10 \):
\[
5 + 5 = 10 \quad \text{(True)}
\]
2. Check the second equation \( 2x - y = 5 \):
\[
2(5) - 5 = 10 - 5 = 5 \quad \text{(True)}
\]
Both equations are satisfied, so the solution is correct.
---
Final Answer:
The solution to the system of equations is:
\[
\boxed{x = 5, y = 5}
\]
Parent Tip: Review the logic above to help your child master the concept of operations with polynomials worksheet.
