Problem 197 - Elementary Algebra

Question

Alyssa is twelve years older than her sister, Bethany. The sum of their ages is forty-four. Find their ages.

Step-by-Step Solution

Answer

Alyssa is 28 and Bethany is 16.

Step 1: Define the Variables

Let Alyssa's age be represented by the variable a , and Bethany's age be represented by the variable b . Form the equations based on the given information.

Step 2: Set Up the Equations

Based on the problem, Alyssa is twelve years older than Bethany, so we can write the equation a = b + 12 . Also, the sum of their ages is forty-four, giving us the equation a + b = 44 .

Step 3: Substitute and Simplify

Substitute a = b + 12 into the equation a + b = 44 to get (b + 12) + b = 44 . Combine like terms to get 2b + 12 = 44 .

Step 4: Solve for Bethany's Age

Subtract 12 from both sides to get 2b = 32 . Then divide both sides by 2 to find b = 16 .

Step 5: Solve for Alyssa's Age

Use the value of b (Bethany's age) in the equation a = b + 12 to find a = 16 + 12 , which gives a = 28 .

Step 6: Verify the Solution

Check that the sum of their ages is forty-four. Substituting the values a = 28 and b = 16 into the equation a + b = 44 confirms 28 + 16 = 44 .

Key Concepts

Linear Equations

Linear equations are mathematical statements that show the equality of two expressions and involve one or more variables raised to the first power. In our exercise, we used linear equations to express the relationship between Alyssa and Bethany's ages. By defining Alyssa's age as \(a\) and Bethany's age as \(b\), and then forming the equations based on the given information, we could set up a system of linear equations.
For example:

• The equation \(a = b + 12\) shows that Alyssa is twelve years older than Bethany.
• The equation \(a + b = 44\) represents the sum of their ages equaling forty-four.

Substitution Method

The substitution method is a technique for solving systems of equations where one equation is solved for one variable in terms of the other variable, and then this expression is substituted into the other equation. This method helps reduce the system to a single equation with one variable, making it easier to solve.
In our exercise, we used the substitution method as follows:

• First, solve one of the equations for one of the variables. We solved \(a = b + 12\) for \(a\).
• Next, substitute this expression into the second equation \(a + b = 44\). This substitution gives us \((b + 12) + b = 44\).

By simplifying this, we can solve for \(b\) and find Bethany's age.

Solving Equations

Solving equations involves finding the value(s) of the variable(s) that make the equation true. After substitution, our task was to simplify and solve the resulting equation.
We started with:

• Simplify \((b + 12) + b = 44\) to get \2b + 12 = 44\.
• Subtract 12 from both sides: \2b = 32\.
• Divide both sides by 2: \b = 16\.

After finding \(b = 16\), we substitute back to find \(a\): \a = 16 + 12\, so \a = 28\.
Finally, we verify the solution by checking if the sum of their ages equals 44: \28 + 16 = 44\.