Question

a first number plus twice a second number is 12. twice the first number plus the second totals 30. find the numbers. the smaller of the two numbers is \\(\square\\).

Explanation:

Step1: Define variables

Let the first number be \( x \) and the second number be \( y \).
From the problem, we have two equations:
\( x + 2y = 12 \) (Equation 1)
\( 2x + y = 30 \) (Equation 2)

Step2: Solve the system of equations

We can use the elimination method. Multiply Equation 1 by 2:
\( 2x + 4y = 24 \) (Equation 3)
Subtract Equation 2 from Equation 3:
\( (2x + 4y) - (2x + y) = 24 - 30 \)
\( 2x + 4y - 2x - y = -6 \)
\( 3y = -6 \)
\( y = -2 \)

Step3: Find the value of \( x \)

Substitute \( y = -2 \) into Equation 1:
\( x + 2(-2) = 12 \)
\( x - 4 = 12 \)
\( x = 12 + 4 \)
\( x = 16 \)

Step4: Determine the smaller number

We have \( x = 16 \) and \( y = -2 \). Since \( -2 < 16 \), the smaller number is \( -2 \).

Answer:

\(-2\)