Question

Three times a larger number is 30 more than 5 times a smaller number. The sum of the larger number and 5 times the smaller number is 50.

Let x be the smaller number and y be the larger number.

Create a system and graph it to find the solution.

What are the two numbers?

x =
and y =

Answer 1

Let's set up the system of equations based on the problem statement.

Let \( x \) be the smaller number and \( y \) be the larger number.

We can formulate the two equations from the problem statements as follows:

1. **From the first statement**: "Three times a larger number is 30 more than 5 times a smaller number."
\[
3y = 5x + 30
\]

2. **From the second statement**: "The sum of the larger number and 5 times the smaller number is 50."
\[
y + 5x = 50
\]

Now we have a system of equations:
\[
\begin{align*}
1) & \quad 3y = 5x + 30 \\
2) & \quad y + 5x = 50
\end{align*}
\]

### Solve the system of equations:

First, we can solve the second equation for \( y \):
\[
y = 50 - 5x
\]

Now, substitute \( y \) into the first equation:
\[
3(50 - 5x) = 5x + 30
\]

Distributing the \( 3 \):
\[
150 - 15x = 5x + 30
\]

Next, combine like terms by moving all terms involving \( x \) to one side and constants to the other:
\[
150 - 30 = 15x + 5x \\
120 = 20x
\]

To solve for \( x \):
\[
x = \frac{120}{20} = 6
\]

Now substitute \( x = 6 \) back into the equation for \( y \):
\[
y = 50 - 5(6) \\
y = 50 - 30 = 20
\]

### Solution:

The smaller number \( x \) is 6, and the larger number \( y \) is 20.

So, we have:
\[
x = 6 \quad \text{and} \quad y = 20
\]