Question

4-3 exercises
a

1. the sum of a certain number and a second number is -42. the first number minus the second is 52. find the numbers.
2. the sum of two numbers is -63. the first number minus the second is -41. find the numbers.
3. the difference between two numbers is 16. three times the larger number is nine times the smaller. what are the numbers?
4. the difference between two numbers is 11. twice the smaller number plus three times the larger number is 123. what are the numbers?
5. soybean meal is 16% protein and corn meal is 9% protein. how many pounds of each should be mixed together to get a 350-lb mixture that is 12% protein?
6. a chemist has one solution that is 25% acid and a second that is 50% acid. how many liters of each should be mixed to get 10 l of a solution that is 40% acid?
7. one canned juice drink is 15% orange juice and another is 5% orange juice. how many liters of each should be mixed together to get a 10 l solution that is 10% orange juice?

Explanation:

Problem 1:

Step1: Define variables

Let the first number be \( x \) and the second number be \( y \).

Step2: Set up equations

From the problem, we have two equations:
\( x + y = -42 \) (Equation 1)
\( x - y = 52 \) (Equation 2)

Step3: Solve the system of equations

Add Equation 1 and Equation 2:
\( (x + y) + (x - y) = -42 + 52 \)
\( 2x = 10 \)
\( x = 5 \)
Substitute \( x = 5 \) into Equation 1:
\( 5 + y = -42 \)
\( y = -42 - 5 \)
\( y = -47 \)

Step1: Define variables

Let the first number be \( x \) and the second number be \( y \).

Step2: Set up equations

We have:
\( x + y = -63 \) (Equation 1)
\( x - y = -41 \) (Equation 2)

Step3: Solve the system

Add Equation 1 and Equation 2:
\( (x + y) + (x - y) = -63 + (-41) \)
\( 2x = -104 \)
\( x = -52 \)
Substitute \( x = -52 \) into Equation 1:
\( -52 + y = -63 \)
\( y = -63 + 52 \)
\( y = -11 \)

Step1: Define variables

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

Step2: Set up equations

We know:
\( x - y = 16 \) (Equation 1, since difference is 16)
\( 3x = 9y \) (Equation 2, three times larger is nine times smaller)
Simplify Equation 2: \( x = 3y \)

Step3: Solve the system

Substitute \( x = 3y \) into Equation 1:
\( 3y - y = 16 \)
\( 2y = 16 \)
\( y = 8 \)
Then \( x = 3\times8 = 24 \)

Answer:

The first number is 5 and the second number is -47.

Problem 2: