Problem 211 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. The difference of two supplementary angles is 8 degrees. Find the measures of the angles.

Step-by-Step Solution

Answer

The angles are 94 degrees and 86 degrees.

Step 1 Title - Define Variables

Let the two angles be represented by the variables \(x\) and \(y\). Supplementary angles add up to 180 degrees, so \(x + y = 180\).

Step 2 Title - Set up the second equation

According to the problem, the difference between the two angles is 8 degrees, so the second equation is \(x - y = 8\).

Step 3 Title - Rewrite the System of Equations

We now have the following system of equations: 1. \(x + y = 180\) 2. \(x - y = 8\)

Step 4 Title - Solve for one variable

Add the two equations together to eliminate \(y\): \((x + y) + (x - y) = 180 + 8\). This simplifies to \(2x = 188\). Divide by 2: \(x = 94\).

Step 5 Title - Substitute to find the second variable

Substitute \(x = 94\) back into the first equation: \(94 + y = 180\). Solving for \(y\): \(y = 180 - 94\) \(y = 86\).

Step 6 Title - Verify the solution

Check the values by substituting them back into the second equation: \(94 - 86 = 8\). Both conditions are satisfied, so the solution is correct.

Key Concepts

Supplementary Angles

In geometry, supplementary angles are two angles whose measures add up to 180 degrees. This relationship is particularly useful in solving various problems in algebra and geometry.
When you are given a pair of angles and asked to find their measures, knowing that they are supplementary can help you set up an equation. For example, if angle 1 is represented by variable x and angle 2 by variable y, then according to the definition of supplementary angles, you have:
\[ x + y = 180 \]
This foundational concept is crucial in solving our example exercise.

Algebraic Equations

Algebraic equations are mathematical expressions that contain variables and constants linked by an equals sign. In our example exercise, we set up a system of algebraic equations to find the values of unknown angles.
Firstly, we used the property of supplementary angles to write our first equation:
\[ x + y = 180 \]
Then, we used the given condition that the difference between the angles is 8 degrees to set up our second equation:
\[ x - y = 8 \]
Together, these two equations form a system of equations. Solving such a system involves finding the values of x and y that satisfy both equations simultaneously.

Solving Equations

Once the system of equations is set up, we proceed to solve it step-by-step. Here's how we solve it in our example:
1. **Rewrite the System of Equations:** We have:
\[\begin{align*} 1. & \, x + y = 180 \ 2. & \, x - y = 8 \right)\right\end{align*}\]2. **Eliminate one variable:** Add the two equations to eliminate y:
\[ (x + y) + (x - y) = 180 + 8 \]This simplifies to:
\[ 2x = 188 \]Divide by 2 to solve for x:
\[ x = 94 \]
3. **Substitute to find the second variable:** Use x = 94 in the first equation:
\[ 94 + y = 180 \]Solving for y:
\[ y = 180 - 94 \]y = 86
4. **Verify the solution:** Substitute back into the second equation to ensure it holds true:
\[ 94 - 86 = 8 \]
Since both conditions are satisfied, the solution \( x = 94 \) and \( y = 86 \) is correct.