Problem 208 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. The difference of two complementary angles is 68 degrees. Find the measures of the angles.

Step-by-Step Solution

Answer

The angles are 79 degrees and 11 degrees.

Step 1: Identify Variables

Let \( x \) and \( y \) represent the measures of the two complementary angles.

Step 2: Set Up Equations

Since the angles are complementary, their sum is 90 degrees. So, the first equation is: \[ x + y = 90 \]

Step 3: Use Given Condition

The problem states that the difference between the two angles is 68 degrees. So, the second equation is: \[ x - y = 68 \]

Step 4: Solve the System of Equations - Add Equations

Add the two equations to eliminate \(y\): \[ (x + y) + (x - y) = 90 + 68 \] Which simplifies to: \[ 2x = 158 \] Now solve for \( x \): \[ x = \frac{158}{2} = 79 \]

Step 5: Solve for the Second Angle

Substitute \( x = 79 \) back into the first equation to find \( y \): \( 79 + y = 90 \) Solve for \( y \): \[ y = 90 - 79 = 11 \]

Step 6: Verify the Solution

Ensure that the difference is 68 degrees: \( 79 - 11 = 68 \) Both conditions are met, so the solution is correct.

Key Concepts

system of equations

A system of equations is a set of equations with multiple variables. In solving a system, we look for common solutions that satisfy all equations simultaneously. This exercise on complementary angles requires us to set up and solve a system of linear equations. In this case, we had two unknowns, the measures of the angles. We developed two equations based on their complementary nature and the given difference between them.

complementary angles

Complementary angles are two angles whose measures add up to 90 degrees. This is a key concept when solving problems involving such angles. In this exercise, we used this property to set up the equation. For instance, if one angle is x degrees, the other would be (90 - x) degrees. This relationship forms the basis of one of our linear equations in the system.

solving linear equations

Solving linear equations involves finding the values of the variables that make the equations true. In this exercise, we solved linear equations both individually (like x + y = 90) and by combining them to eliminate a variable. When we added our two equations, we eliminated one variable, making it simpler to solve for the remaining variable. This method is often called the elimination method.

algebra step-by-step

Step-by-step solving in algebra ensures clarity and understanding. Let's revisit the steps we used:

• Identify variables: Let x and y be the angles.
• Set up equations: Use complementary angle sum and given difference.
• Solve the system: Add to eliminate y, solve for x, then find y.
• Verify solution: Ensure both initial conditions are met.

This structured approach allows us to methodically solve more complex problems by breaking them down into manageable parts.