Chapter2: Equations And Inequalities
Section: Chapter Questions
Problem 4PT: Find the exact distance between (5,3)and(2,8) . Find the coordinates of the midpoint of the line...
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Solving for X and Y
![## Problem Statement:
Find the values of \(x\) and \(y\).
### Diagram Explanation:
This is an angle relationship diagram with the following components:
- There are five points labeled \(V\), \(U\), \(R\), \(S\), and \(T\).
- These points are connected by lines forming several angles around point \(R\).
- There are three angles with given expressions:
- The angle \(\angle VRS\) is labeled as \((12x - 4)^\circ\).
- The angle \(\angle URS\) is labeled as \((4y)^\circ\).
- The angle \(\angle URT\) is labeled as \((5y)^\circ\).
- The angle \( \angle SRU \) is a right angle, meaning it is \(90^\circ\).
### Steps to Solve:
1. Notice that angle \(\angle URS\) and angle \(\angle URT\) together form a straight line, hence their sum is \(180^\circ\):
\[
\angle URS + \angle URT = 180^\circ
\]
Substituting the given expressions:
\[
4y + 5y = 180^\circ \implies 9y = 180^\circ \implies y = 20^\circ
\]
2. Considering the right angle relation between \(\angle URS\) and \(\angle TRS\), we know:
\[
\angle URS + \angle TRS = 90^\circ
\]
Substituting \(\angle URS\) as \(4y\):
\[
4y + \angle TRS = 90^\circ \implies 4(20) + \angle TRS = 90^\circ \implies 80 + \angle TRS = 90^\circ \implies \angle TRS = 10^\circ
\]
3. Next, consider the overall linear angle \(\angle VRS + \angle URS + \angle TRS = 180^\circ\):
\[
(12x-4) + 4y + \angle TRS = 180^\circ
\]
Substituting the known values:
\[
(12x - 4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5bd50398-42a6-49cb-bcb5-428df3415a3d%2F4488cad9-c452-4e77-8355-1e8f09dd6014%2Fx44h0c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Problem Statement:
Find the values of \(x\) and \(y\).
### Diagram Explanation:
This is an angle relationship diagram with the following components:
- There are five points labeled \(V\), \(U\), \(R\), \(S\), and \(T\).
- These points are connected by lines forming several angles around point \(R\).
- There are three angles with given expressions:
- The angle \(\angle VRS\) is labeled as \((12x - 4)^\circ\).
- The angle \(\angle URS\) is labeled as \((4y)^\circ\).
- The angle \(\angle URT\) is labeled as \((5y)^\circ\).
- The angle \( \angle SRU \) is a right angle, meaning it is \(90^\circ\).
### Steps to Solve:
1. Notice that angle \(\angle URS\) and angle \(\angle URT\) together form a straight line, hence their sum is \(180^\circ\):
\[
\angle URS + \angle URT = 180^\circ
\]
Substituting the given expressions:
\[
4y + 5y = 180^\circ \implies 9y = 180^\circ \implies y = 20^\circ
\]
2. Considering the right angle relation between \(\angle URS\) and \(\angle TRS\), we know:
\[
\angle URS + \angle TRS = 90^\circ
\]
Substituting \(\angle URS\) as \(4y\):
\[
4y + \angle TRS = 90^\circ \implies 4(20) + \angle TRS = 90^\circ \implies 80 + \angle TRS = 90^\circ \implies \angle TRS = 10^\circ
\]
3. Next, consider the overall linear angle \(\angle VRS + \angle URS + \angle TRS = 180^\circ\):
\[
(12x-4) + 4y + \angle TRS = 180^\circ
\]
Substituting the known values:
\[
(12x - 4
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