Find the solution of the system of equations.
6, x, plus, 2, y, equals, 16
6x+2y=
16
2, x, minus, 2, y, equals, 32
2x−2y=
32

To solve the system of equations:

\[
6x + 2y = 16
\]
\[
2x - 2y = 32
\]

we can use either substitution or elimination. Here, elimination looks straightforward, so let's proceed with that.

### Step 1: Simplify the Equations
First, divide the first equation by 2 for easier manipulation:
\[
3x + y = 8
\]

Now the system becomes:
1. \(3x + y = 8\)
2. \(2x - 2y = 32\)

### Step 2: Eliminate \(y\) by Adding the Equations
To eliminate \(y\), let’s manipulate the second equation. Divide the second equation by 2 to simplify it:
\[
x - y = 16
\]

Now the system is:
1. \(3x + y = 8\)
2. \(x - y = 16\)

Add these two equations:
\[
(3x + y) + (x - y) = 8 + 16
\]
\[
4x = 24
\]

### Step 3: Solve for \(x\)
Divide both sides by 4:
\[
x = 6
\]

### Step 4: Substitute \(x\) into One of the Equations to Find \(y\)
Now, substitute \(x = 6\) into the simplified equation \(x - y = 16\):
\[
6 - y = 16
\]
Subtract 6 from both sides:
\[
-y = 10
\]
Multiply by \(-1\):
\[
y = -10
\]

### Solution
The solution to the system is:
\[
x = 6, \quad y = -10
\]

Would you like more details or have any questions?

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Here are five related questions to explore further:
1. What method would you use to solve a system with three variables?
2. How can you verify that your solution to a system of equations is correct?
3. What does it mean if a system of equations has no solution?
4. Can a system of equations have infinitely many solutions, and if so, how is that identified?
5. How does graphing the equations help in understanding their solutions?

**Tip:** When using elimination, simplifying equations by dividing can make calculations easier, as shown above.

Find the solution of the system of equations: 6x + 2y = 16, 2x - 2y = 32.

Explore how to solve the system of equations 6x + 2y = 16 and 2x - 2y = 32 using the elimination method. This solution covers core algebraic concepts and demonstrates step-by-step simplification and substitution, making it ideal for Grades 8-10.

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