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 -2.
the larger 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 the following two equations:
\( x + 2y=12 \) (Equation 1)
\( 2x + y = 30 \) (Equation 2)

Step2: Solve the system of equations

We can use the elimination method. First, multiply Equation 1 by 2:
\( 2(x + 2y)=2\times12 \)
\( 2x+4y = 24 \) (Equation 3)

Now subtract Equation 2 from Equation 3:
\( (2x + 4y)-(2x + y)=24 - 30 \)
\( 2x+4y - 2x - y=- 6 \)
\( 3y=-6 \)
Divide both sides by 3:
\( y=\frac{-6}{3}=-2 \)

Now substitute \( y = - 2 \) into Equation 1:
\( x+2\times(-2)=12 \)
\( x - 4=12 \)
Add 4 to both sides:
\( x=12 + 4=16 \)

Answer:

The larger of the two numbers is \( 16 \).