Solve the system of equations x+y = 5 and x−y = 1. What are the values of x and y?


To solve the system of equations:

\[
\begin{aligned}
1) & \quad x + y = 5 \\
2) & \quad x - y = 1
\end{aligned}
\]

We can use the method of addition (also known as elimination) to find the values of \(x\) and \(y\).

### Step 1: Add the two equations
By adding the two equations, we can eliminate \(y\):

\[
(x + y) + (x - y) = 5 + 1
\]

This simplifies to:

\[
2x = 6
\]

### Step 2: Solve for \(x\)
Now, divide both sides by 2:

\[
x = \frac{6}{2} = 3
\]

### Step 3: Substitute \(x\) back into one of the original equations
We'll substitute \(x = 3\) into the first equation \(x + y = 5\):

\[
3 + y = 5
\]

### Step 4: Solve for \(y\)
Subtract 3 from both sides:

\[
y = 5 - 3 = 2
\]

### Final Answer:
The values of \(x\) and \(y\) are:

\[
x = 3, \quad y = 2
\]

Would you like further details or have any questions?

Here are some related questions:
1. How would you solve the system \(2x + y = 7\) and \(x - y = 4\)?
2. What is the substitution method for solving systems of equations?
3. How can you solve a system of equations using matrices?
4. Can you explain the graphical interpretation of solving a system of linear equations?
5. What happens if a system of equations has no solution?

**Tip:** When solving systems of equations, always check your solution by substituting the values back into both original equations.

Learn how to solve the system of equations x + y = 5 and x - y = 1 with a step-by-step solution. This problem covers key concepts in systems of linear equations and methods such as addition (elimination) and substitution. Suitable for students in Grades 7-9.

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