Problem 214 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. The difference of two complementary angles is 17 degrees. Find the measures of the angles.

Step-by-Step Solution

Answer

The angles are 53.5 degrees and 36.5 degrees.

Step 1 - Define the Variables

Let the two complementary angles be represented by \(x\) and \(y\).

Step 2 - Set Up the Equations

Complementary angles sum up to 90 degrees. Therefore, we have:\[x + y = 90\] The problem also states that the difference between the angles is 17 degrees. Therefore, we have another equation:\[x - y = 17\]

Step 3 - Solve the System of Equations

To solve the system, add the two equations:\[(x + y) + (x - y) = 90 + 17\]\[2x = 107\]\[x = 53.5\]Now substitute \(x = 53.5\) back into the first equation:\[53.5 + y = 90\]Solve for \(y\):\[y = 90 - 53.5 = 36.5\]

Step 4 - Verify the Solution

Check that the sum of the angles is 90 degrees:\[53.5 + 36.5 = 90\]Also, confirm their difference is 17 degrees:\[53.5 - 36.5 = 17\]

Key Concepts

System of Equations

When solving problems involving relationships between multiple unknowns, we often use a **system of equations**. A system of equations is a set of two or more equations with the same variables. Our goal is to find the values of these variables that satisfy all the equations in the system simultaneously.
In this exercise, we need to find two angles that not only add up to 90 degrees (complementary) but also differ by 17 degrees. Therefore, our system of equations is:
\[x + y = 90\] \[x - y = 17\]
Each equation expresses a different relationship between the variables. By solving this system, we can find the precise values of the angles involved in the problem.

Angle Measures

Angles are measured in degrees, and **complementary angles** are two angles whose measures add up to 90 degrees. Understanding this concept is crucial because it provides the foundation for setting up the system of equations in this problem.
Here are some key things to remember about angles:

• **Complementary angles** sum up to 90 degrees.
• **Supplementary angles** sum up to 180 degrees.
• The **difference** between angles provides another useful equation.

Using these properties, we defined two complementary angles, represented them algebraically with variables, and set up two related equations. In this specific exercise, knowing the angles are complementary led us to the equation \[x + y = 90\], while the description of their difference led us to \[x - y = 17\].

Algebraic Solutions

To find the values of the unknowns in our system of equations, we use algebraic methods. This involves combining and manipulating equations to isolate each variable.
In this problem, we followed these steps:

• **Adding the equations**: We combined \[x + y = 90\] and \[x - y = 17\] to eliminate variable 'y'. This gave us \[2x = 107\].
• **Solving for x**: Next, we divided both sides of \[2x = 107\] by 2 to find \[x = 53.5\].
• **Substituting back**: We then substituted \[x = 53.5\] into the first equation to solve for 'y', leading to \[y = 36.5\].

Finally, we **verified the solution** by checking:

• The sum of the angles: \[53.5 + 36.5 = 90\].
• The difference of the angles: \[53.5 - 36.5 = 17\].

Both verifications confirmed the correctness of our solution.