Math Problem Statement

Given: ∠TRS is a straight angle, ∠TRX is a right angle, m∠TRS = 2x + 5y, and m∠XRS = 3x + 3y. Solve for x and y.

Solution

The problem provides the following information:

  1. ∠TRS is a straight angle, which means that its measure is 180°.
  2. ∠TRX is a right angle, which means that its measure is 90°.
  3. The measures of the angles are given as:
    • mTRS=2x+5ym\angle TRS = 2x + 5y
    • mXRS=3x+3ym\angle XRS = 3x + 3y

We need to solve for xx and yy.

Solution:

Since TRS\angle TRS is a straight angle, it can be divided into two parts: TRX\angle TRX and XRS\angle XRS.

TRS=TRX+XRS\angle TRS = \angle TRX + \angle XRS

Substitute the given values:

2x+5y=90+(3x+3y)2x + 5y = 90 + (3x + 3y)

Simplifying:

2x+5y=90+3x+3y2x + 5y = 90 + 3x + 3y

Rearrange to solve for xx and yy:

2x+5y3x3y=902x + 5y - 3x - 3y = 90 x+2y=90-x + 2y = 90 x=2y90(Equation 1)x = 2y - 90 \quad \text{(Equation 1)}

Next, we substitute this value of xx into the given mTRX=90°m\angle TRX = 90° and verify.

Check the value:

We substitute x=2y90x = 2y - 90 into original equation:

Result...

Let's calculate directly.The solution to the equations is:

x=2y90x = 2y - 90

This means xx is expressed in terms of yy. There isn't a unique solution for both xx and yy since we need an additional independent equation to solve for both variables.

Would you like to discuss this further or solve additional problems related to this topic?

Related Questions

  1. If y=50y = 50, what is the value of xx?
  2. How do you find the complementary angles to TRX\angle TRX and XRS\angle XRS?
  3. What are the conditions for two lines to form a straight angle?
  4. Can two right angles form a straight line?
  5. How do angle measures change with different xx and yy values?

Tip: Always check if the number of equations matches the number of variables when solving a system of linear equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Linear Equations
Angle Relationships

Formulas

Sum of angles on a straight line: ∠TRS = 180°
Right angle property: ∠TRX = 90°

Theorems

Angle Sum Theorem
Linear Pair Theorem

Suitable Grade Level

Grades 9-10