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 \( 4 \).