Problem 60 - Algebra 2

Question

For Exercises \(58-63,\) define a variable, write an equation, and solve the problem. AGES Chun-Wei's mother is 8 more than twice his age. His father is three years older than his mother is. If the three family members have lived a total of 94 years, how old is each family member?

Step-by-Step Solution

Answer

Chun-Wei is 15, his mother is 38, and his father is 41 years old.

Step 1: Define the variables

Let Chun-Wei's age be denoted by \( x \). Thus, we will define the ages of each family member in terms of \( x \). This gives us the following relationships: Chun-Wei's age \( = x \), his mother's age \( = 2x + 8 \), and his father's age \( = 2x + 8 + 3 \).

Step 2: Set up the equation

The problem states that the sum of the ages of the three family members is 94 years. Using our definitions, the equation becomes: \[ x + (2x + 8) + (2x + 8 + 3) = 94 \].

Step 3: Simplify the equation

Combine like terms in the equation: \[ x + 2x + 8 + 2x + 8 + 3 = 94 \] simplifies to \[ 5x + 19 = 94 \].

Step 4: Solve for \( x \)

Subtract 19 from both sides of the equation to isolate the term with \( x \): \[ 5x = 75 \]. Then, divide both sides by 5 to solve for \( x \): \[ x = 15 \].

Step 5: Determine the ages of each family member

Now that we know \( x = 15 \), substitute back into the age expressions for each family member. Chun-Wei is 15 years old, his mother is \( 2(15) + 8 = 38 \) years old, and his father is \( 38 + 3 = 41 \) years old.

Key Concepts

Defining Variables

When we solve age word problems, the first step is defining variables. A variable acts as a placeholder for unknown information we need to find. In this exercise, we start by letting Chun-Wei's age be the unknown quantity, which we can express as the variable \( x \).
This allows us to write down relationships involving the ages of the other family members. For example:

• Chun-Wei's age is \( x \).
• His mother's age, described as "8 more than twice" his age, is \( 2x + 8 \).
• His father's age, "three years older than his mother," becomes \( 2x + 8 + 3 \) or \( 2x + 11 \).

By defining variables, you create a clear starting point to tackle the problem.

Writing Equations

After defining the variables, the next step is to write an equation based on the information given. An equation is a mathematical statement that shows the equality of two expressions, allowing us to solve for the unknowns.
In this problem, it is given that the total age of the family members is 94. Based on our variable definitions, we can write the equation:

• \( x + (2x + 8) + (2x + 11) = 94 \).

The equation includes all three family members' ages, showing their sum is 94. Writing equations is crucial because it translates the problem's words into a form we can solve.

Solving Equations

Solving the equation is our main goal to find the values of the variables. First, we simplify the equation by combining like terms.
From the equation \( x + (2x + 8) + (2x + 11) = 94 \), we combine the coefficients of \( x \) and constant terms:

• This simplifies to \( 5x + 19 = 94 \).

We then isolate the term containing \( x \) by subtracting 19 from both sides:

• \( 5x = 75 \).

Finally, divide by 5 to find \( x \):

• \( x = 15 \).

The solution provides Chun-Wei's age, enabling us to further calculate his parents' ages.

Linear Equations

Age problems often involve linear equations, which represent straight-line relationships. The main feature of a linear equation is its highest variable exponent is 1. This keeps the calculation straightforward and manageable.
Our equation, \( 5x + 19 = 94 \), exemplifies a linear equation because:

• The variables are simplified to the first power.
• It allows us to predict and find solutions easily by performing arithmetic operations.

Understanding linear equations ensures you can work with various algebraic problems systematically, simplifying the path to finding solutions.