Question

The sum of two numbers is 12 and their difference is 4. Define variables, write a system of equations and solve to find the two numbers.

Answer 1

Let's define the two numbers:

Let \( x \) be the first number and \( y \) be the second number.

From the problem, we have two pieces of information that we can turn into equations:

1. The sum of the two numbers is 12.
\[
x + y = 12
\]
2. The difference between the two numbers is 4.
\[
x - y = 4
\]

Now we have a system of equations:
\[
\begin{align*}
1) & \quad x + y = 12 \\
2) & \quad x - y = 4
\end{align*}
\]

Next, we can solve this system of equations. We can use the method of substitution or elimination. Here, we'll use elimination.

First, let's add the two equations together to eliminate \( y \):
\[
(x + y) + (x - y) = 12 + 4
\]
This simplifies to:
\[
2x = 16
\]
Now, we can solve for \( x \) by dividing both sides by 2:
\[
x = 8
\]

Now that we have \( x \), we can substitute it back into one of the original equations to solve for \( y \). We'll use the first equation:
\[
8 + y = 12
\]
Subtracting 8 from both sides gives:
\[
y = 4
\]

So, the two numbers are:
\[
x = 8 \quad \text{and} \quad y = 4
\]

To summarize, the solution is:
- The first number is \( 8 \).
- The second number is \( 4 \).