Question

solve the following by setting up the proper equations to describe the facts given and then carrying out the mathematical calculations to solve for the unknown variable(s). let x represent the larger number and y represent the smaller number. enter your answer in (x, y) format.

7 times a larger number added to 16 times a smaller number totals 74. when 4 times the smaller number is subtracted from 2 times the larger number, the result is 4. what are the two numbers?

Explanation:

Translate the word problem into equations

Using the Translating Word Problems knowledge point
Let \(x\) be the larger number and \(y\) be the smaller number.
The first statement: "7 times a larger number added to 16 times a smaller number totals 74"

$$7x + 16y = 74$$

The second statement: "When 4 times the smaller number is subtracted from 2 times the larger number, the result is 4"

$$2x - 4y = 4$$

Solve the system of linear equations

Using the Systems of Linear Equations knowledge point
We have the system:

$$
LATEXBLOCK0
$$

Multiply Equation 2 by 4:

$$8x - 16y = 16 \quad \text{(Equation 3)}$$

Add Equation 1 and Equation 3:

$$
LATEXBLOCK1
$$

Substitute \(x = 6\) into Equation 2:

$$
LATEXBLOCK2
$$

The solution is \((6, 2)\).

Answer:

Solve the following by setting up the proper equations to describe the facts given and then carrying out the mathematical calculations to solve for the unknown variable(s). Let x represent the larger number and y represent the smaller number. Enter your answer in (x, y) format.

7 times a larger number added to 16 times a smaller number totals 74. When 4 times the smaller number is subtracted from 2 times the larger number, the result is 4. What are the two numbers? <blank>\((6, 2)\)</blank>