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:
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From the first statement: "Three times a larger number is 30 more than 5 times a smaller number." \[ 3y = 5x + 30 \]
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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: \[ \begin{align*}
- & \quad 3y = 5x + 30 \
- & \quad y + 5x = 50 \end{align*} \]
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: \[ x = 6 \quad \text{and} \quad y = 20 \]
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