Question

Lisa and Erica were investigating the relationship of their ages. They figured out that Erica’s age is currently 5 less than twice Lisa’s age. In addition, the sum of their ages is 73. How old are Erica and Lisa? (A) Erica is 47, and Lisa is 26. (B) Erica is 26, and Lisa is 47. (C) Erica is 20, and Lisa is 53. (D) There is not enough information to determine the answer.

Short answer

(A) Erica is 47, and Lisa is 26.

Step-by-step solution

Define the variables

Let E be Erica's age and L be Lisa's age currently.

Write the given relationships as equations

The problem states that Erica's age (E) is 5 less than twice Lisa's age (L), so we have: \( E = 2L - 5 \) The problem also states that the sum of their ages (E + L) is 73, so we have: \( E + L = 73 \)

Solve the system of equations

We will use the substitution method to solve these equations. We can substitute the equation for E (from Step 2) into the equation for the sum of their ages, so we get: \( (2L - 5) + L = 73 \)

Simplify and solve for L

Combine the L terms and solve for L: \( 3L - 5 = 73 \) Add 5 to both sides of the equation to isolate the term with L: \( 3L = 78 \) Now, divide by 3 to find L: \( L = 26 \) So, Lisa is 26 years old.

Find Erica's age

Now that we know Lisa is 26, we can use the equation from step 2 to find Erica's age: \( E = 2L - 5 \) Substitute 26 for L: \( E = 2(26) - 5 \) Simplify the equation: \( E = 52 - 5 \) \( E = 47 \) So, Erica is 47 years old.

Interpret the solution

Using the given relationships, we found that Lisa is currently 26 years old, and Erica is 47 years old. Thus, the correct answer is (A) Erica is 47, and Lisa is 26.

Key concepts

System of Equations

In algebra, age-related problems often involve figuring out relationships between different ages using equations. We use a **system of equations** when there is more than one equation involving multiple variables that we need to solve simultaneously. In our exercise, we have two equations involving the ages of Erica and Lisa.

One equation states that Erica's age is 5 less than twice Lisa's age, represented as:

• \( E = 2L - 5 \)

The second equation states that the sum of Erica and Lisa's ages is 73:

• \( E + L = 73 \)

To solve these equations, we need to find values for the variables (their ages) that satisfy both. This is achieved by using different methods such as substitution or elimination. Such problems simulate real-life situations where you balance multiple conditions to find a solution that works for all.

Substitution Method

The **substitution method** is a way to solve a system of equations by expressing one variable in terms of another. This method is very useful when dealing with age problems, as it can simplify the equations involved. In our specific task, since we know the relationship between Erica and Lisa's ages (\( E = 2L - 5 \)), we can substitute this expression into another equation.

Let’s substitute \( E \) in the equation representing the sum of their ages:

• Substitute \( E = 2L - 5 \) into \( E + L = 73 \) gives:
• \((2L - 5) + L = 73\)

This substitution transforms the system of equations into a single equation with one variable, making it much easier to solve. By combining terms and isolating the variable, we can solve for Lisa's age, then use that value to find Erica's age.

The substitution method is particularly powerful when one of the equations is already solved for a variable, as it reduces the complexity and helps in finding the solution efficiently.

Variables in Algebra

In algebra, **variables** are symbols used to represent numbers whose values are not yet known. They are often represented by letters like \( x \), \( y \), \( E \), or \( L \). In our age problem, we defined the variables as:

• \( E \): Erica’s age
• \( L \): Lisa’s age

By defining these variables, we transformed a word problem into a mathematical model that is easier to work with.

Using variables allows us to write equations that describe the relationships and constraints provided in the problem. This mathematical model can then be manipulated using algebraic techniques to find solutions.

When working with variables in such contexts, it's essential to carefully define what each represents and maintain consistency throughout the calculations. This helps ensure that we interpret the solutions correctly in the context of the original problem.