Certainly! Let's go through the reasoning and calculations step-by-step.
1. Understand the angles and their measures:
- ZAPR is a straight angle, so it measures 180 degrees.
- ZAPS is a right angle, so it measures 90 degrees.
- m/APR and m/SPR both contribute to the straight angle ZAPR.
2. Set up the expressions for the angles:
- m/APR is given as [tex]\(2x + 5y\)[/tex].
- m/SPR is given as [tex]\(3x + 3y\)[/tex].
3. Set up the equations based on angle properties:
- Since ZAPR is a straight angle:
[tex]\( \text{m/APR} + \text{m/SPR} = 180^\circ \)[/tex]
Substituting the given expressions:
[tex]\[ (2x + 5y) + (3x + 3y) = 180 \][/tex]
Simplifying this, we get:
[tex]\[ 5x + 8y = 180 \][/tex]
This is our first equation (1).
- Since ZAPS is a right angle:
[tex]\[ \text{m/ZAPS} = 90^\circ \][/tex]
Given that m/APR contributes part of ZAPS:
[tex]\[ 2x + 5y = 90 \][/tex]
This is our second equation (2).
4. Solve the system of equations:
We now have the following system of linear equations:
[tex]\[ 5x + 8y = 180 \quad \text{(Equation 1)} \][/tex]
[tex]\[ 2x + 5y = 90 \quad \text{(Equation 2)} \][/tex]
- To solve this system, we'll use the elimination method. First, let's multiply the second equation by 5 and the first equation by 2 to align the coefficients of [tex]\(y\)[/tex]:
[tex]\[ 10x + 25y = 450 \quad \text{(5 multiplied with Equation 2)} \][/tex]
[tex]\[ 10x + 16y = 360 \quad \text{(2 multiplied with Equation 1)} \][/tex]
- Next, subtract the first modified equation from the second modified equation:
[tex]\[ (10x + 25y) - (10x + 16y) = 450 - 360 \][/tex]
[tex]\[ 9y = 90 \][/tex]
[tex]\[ y = 10 \][/tex]
- Substitute [tex]\( y = 10 \)[/tex] back into Equation 2 to find [tex]\( x \)[/tex]:
[tex]\[ 2x + 5(10) = 90 \][/tex]
[tex]\[ 2x + 50 = 90 \][/tex]
[tex]\[ 2x = 40 \][/tex]
[tex]\[ x = 20 \][/tex]
5. Conclusion:
The values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are:
[tex]\[ x = 20 \][/tex]
[tex]\[ y = 10 \][/tex]
So, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are [tex]\(20\)[/tex] and [tex]\(10\)[/tex] respectively.
