Problem 217 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. Two angles are complementary. The measure of the larger angle is twelve less than twice the measure of the smaller angle. Find the measures of both angles.

Step-by-Step Solution

Answer

The smaller angle is 34 degrees and the larger angle is 56 degrees.

Step 1: Understand the Problem

Two angles are complementary, meaning their measures add up to 90 degrees. Let's denote the smaller angle as \( x \) and the larger angle as \( y \).

Step 2: Set Up the Equations

From the problem, we know:1. The sum of the angles is 90 degrees: \( x + y = 90 \).2. The larger angle is 12 less than twice the smaller angle: \( y = 2x - 12 \).

Step 3: Substitute the Second Equation into the First

Substitute \( y = 2x - 12 \) into \( x + y = 90 \) to get:\( x + (2x - 12) = 90 \)

Step 4: Solve for \( x \)

Combine like terms and solve for \( x \):\( x + 2x - 12 = 90 \)\( 3x - 12 = 90 \)Add 12 to both sides:\( 3x = 102 \)Divide by 3:\( x = 34 \)

Step 5: Find \( y \) Using the Value of \( x \)

Now that \( x \) is known, use \( y = 2x - 12 \) to find \( y \):\( y = 2(34) - 12 \)\( y = 68 - 12 \)\( y = 56 \)

Step 6: Verify the Solution

Check that the angles are complementary:\( x + y = 34 + 56 = 90 \). They add up to 90 degrees, so the solution is correct.

Key Concepts

System of Equations

To solve real-world problems involving unknown values, we frequently use systems of equations. Here, we had two unknown angles defined by certain relationships. We formulated two equations:

• Sum of the angles: \(x + y = 90\)
• Larger angle's relationship to the smaller: \(y = 2x - 12\)

With these equations, we created a 'system of equations.' A system of equations consists of two or more equations with the same set of unknowns. In this case, our unknowns are the measures of the two angles. We solve systems using different methods including substitution, elimination, and graphical methods.
We chose the substitution method for this problem, where we solve one of the equations for one variable and then substitute that expression into the second equation.

Angle Relationships

Angles can have various specific relationships with each other, making it possible to solve for their measures using algebra. Here, we dealt with complementary angles:
- **Complementary Angles**: Two angles are complementary if the sum of their measures is 90 degrees. This is the basis of our first equation: \(x + y = 90\).
Understanding angle relationships helps in setting up the correct equations to solve a problem. Complementary, supplementary, and vertical angles are common angle relationships studied in geometry. For this exercise, recognizing the angles as complementary allowed us to determine that their sum must be 90 degrees.

Algebraic Substitution

Algebraic substitution is an effective method for solving systems of equations. It involves substituting one equation into another to eliminate one variable, allowing us to solve for the other.
In our problem, we had:

• \(y = 2x - 12 \) (from the relationship between larger and smaller angle)
• \(x + y = 90 \) (from the complementary angle relationship)

We substituted \(y \) in the first equation into the second:
\ x + (2x - 12) = 90 \
By simplifying and solving this equation, we found the value of \(x\). Then, we used this value to find \(y\) by substituting back into the first equation. Substitution ensures we maintain the equations' relationships and find the exact solution:

1. Solve for one variable in terms of the other: \(y = 2x - 12 \)
2. Substitute this into the other equation to create a single-variable equation: \(x + (2x - 12) = 90\)
3. Solve the single-variable equation: \(3x - 12 = 90\)
4. Use the solution to find the remaining variable: \(x = 34\), \(y = 56\)

This method makes solving the system straightforward and structured.