Problem 191 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve the system. A married couple together earn \(\$ 75,000\). The husband earns \(\$ 15,000\) more than five times what his wife earns. What does the wife earn?

Step-by-Step Solution

Answer

The wife earns \(\$10,000\).

Step 1: Define Variables

Let \(H\) represent the husband's earnings and \(W\) represent the wife's earnings.

Step 2: Formulate Equations

Using the given information, we can write the following two equations: 1. \(H + W = 75000\) 2. \(H = 5W + 15000\)

Step 3: Substitute the Second Equation into the First

Replace \(H\) in the first equation with the expression from the second equation:\(5W + 15000 + W = 75000\)

Step 4: Combine Like Terms

Combine the \(W\) terms in the equation:\(6W + 15000 = 75000\)

Step 5: Isolate \(W\)

Subtract 15000 from both sides of the equation:\(6W = 60000\)Next, divide both sides by 6 to find \(W\):\(W = 10000\)

Step 6: Verify Solution

Substitute \(W = 10000\) back into both original equations to ensure the solution is correct. Using equation 2: \(H = 5(10000) + 15000 = 65000\). Using equation 1: \(65000 + 10000 = 75000\), which is correct.

Key Concepts

variables

In algebra, variables are symbols that represent unknown values. For example, in the problem about the married couple's earnings, we use variables to represent the husband's and wife's earnings. We define these as:
Let H represent the husband's earnings.
Let W represent the wife's earnings.
Choosing appropriate variables helps us to translate real-world problems into mathematical equations that we can solve.

substitution method

The substitution method is a technique for solving systems of equations. This method involves substituting one equation into another to eliminate one of the variables. For this problem, we had two equations:

• 1. H + W = 75000
• 2. H = 5W + 15000

We can substitute the expression for H from the second equation into the first equation. This substitution helps us to solve for W because it gives us one equation with one variable. Thus, the problem becomes easier.

linear equations

Linear equations are equations where the variables are not raised to any power other than one. They graph as straight lines on a coordinate plane. Our problem involved two linear equations:
H + W = 75000 and H = 5W + 15000.
These equations are linear because the variables H and W are to the first power. By solving these linear equations, we can determine the values of the husband's and wife's earnings.

solving equations

Solving equations involves finding the values of the variables that make the equation true. In this problem, after substituting H from the second equation into the first equation, we combined like terms to simplify the equation:
6W + 15000 = 75000.
Next, we isolated W by subtracting 15000 from both sides to get:
6W = 60000.
Finally, we divided by 6 to find the wife's earnings:
W = 10000.
To ensure our solution is correct, we checked it by substituting W back into the original equations. The calculations confirmed that both equations were satisfied.