Grant Kirk
03/30/2024 · Primary School
7- In triangle \( P Q R \), the measure of angle \( P \) is \( (3 x+5)^{\circ} \). The measure of angle \( Q \) is \( (2 x+9)^{\circ} \), and the measure of angle \( R \) is \( (4 y+5)^{\circ} \). If side \( Q R \) is extended through point \( R \) to point \( S \), and the measure of angle \( P R S \) is \( (x+y)^{\circ} \). What is the value of \( x+y \) ?
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Step-by-step Solution
In triangle \( PQR \), the sum of the angles is given by:
\[
\angle P + \angle Q + \angle R = 180^\circ
\]
Substituting the expressions for the angles, we have:
\[
(3x + 5) + (2x + 9) + (4y + 5) = 180
\]
Combining like terms:
\[
3x + 2x + 4y + 5 + 9 + 5 = 180
\]
This simplifies to:
\[
5x + 4y + 19 = 180
\]
Subtracting 19 from both sides gives:
\[
5x + 4y = 161 \quad \text{(1)}
\]
Next, we consider the angle \( PRS \). Since \( QR \) is extended to \( S \), we have:
\[
\angle PRS = 180^\circ - \angle R
\]
Substituting the expression for \( \angle R \):
\[
\angle PRS = 180^\circ - (4y + 5) = 175 - 4y
\]
We are given that:
\[
\angle PRS = (x + y)^\circ
\]
Setting the two expressions for \( \angle PRS \) equal to each other:
\[
175 - 4y = x + y
\]
Rearranging this equation gives:
\[
175 = x + 5y \quad \text{(2)}
\]
Now we have a system of equations:
1. \( 5x + 4y = 161 \)
2. \( x + 5y = 175 \)
We can solve this system using substitution or elimination. Let's solve equation (2) for \( x \):
\[
x = 175 - 5y
\]
Substituting this expression for \( x \) into equation (1):
\[
5(175 - 5y) + 4y = 161
\]
Expanding this gives:
\[
875 - 25y + 4y = 161
\]
Combining like terms results in:
\[
875 - 21y = 161
\]
Subtracting 875 from both sides:
\[
-21y = 161 - 875
\]
\[
-21y = -714
\]
Dividing both sides by -21:
\[
y = \frac{714}{21} = 34
\]
Now substituting \( y = 34 \) back into the expression for \( x \):
\[
x = 175 - 5(34)
\]
\[
x = 175 - 170 = 5
\]
Finally, we find \( x + y \):
\[
x + y = 5 + 34 = 39
\]
Thus, the value of \( x + y \) is:
\[
\boxed{39}
\]
Quick Answer
The value of \( x + y \) is 39.
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