Math Problem Statement

Find the value of each variable and the measure of each angle from the given equations (4x - y)°, 5y°, and (x + y)°.

Solution

In the diagram, we see that the angles formed are vertical angles, which means opposite angles are equal.

From the diagram:

  • (4xy)(4x - y)^\circ and 5y5y^\circ are opposite angles, so: 4xy=5y4x - y = 5y

  • (x+y)(x + y)^\circ is also a vertical angle, so it equals 5y5y^\circ. Thus: x+y=5yx + y = 5y

Step 1: Solve the system of equations

We now have two equations:

  1. 4xy=5y4x - y = 5y
  2. x+y=5yx + y = 5y

Step 2: Solve for yy

From the second equation x+y=5yx + y = 5y, solve for xx in terms of yy: x=5yy=4yx = 5y - y = 4y

Step 3: Substitute x=4yx = 4y into the first equation

Now substitute x=4yx = 4y into the first equation 4xy=5y4x - y = 5y: 4(4y)y=5y4(4y) - y = 5y 16yy=5y16y - y = 5y 15y=5y15y = 5y Subtract 5y5y from both sides: 10y=010y = 0 Thus: y=0y = 0

Step 4: Find xx

Substitute y=0y = 0 back into x=4yx = 4y: x=4(0)=0x = 4(0) = 0

Conclusion

Both xx and yy are 0. Since x=0x = 0 and y=0y = 0, all the angles in the diagram will also measure 0°. However, this result might indicate a need for rechecking the question context or parameters, as having all variables and angles equal to zero could be unusual in this context.

Would you like further clarification or details?

Here are some related questions to explore further:

  1. What are vertical angles, and why are they equal?
  2. How do you solve systems of equations with two variables?
  3. Why might the result of x=0x = 0 and y=0y = 0 suggest a potential issue?
  4. Can geometric problems have trivial or zero solutions?
  5. How do linear equations behave when multiple variables are involved?

Tip: Always check your solution back in the original problem to verify that the result makes sense within the context of the problem.

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Math Problem Analysis

Mathematical Concepts

Algebra
Angle Properties
System of Equations

Formulas

4x - y = 5y
x + y = 5y

Theorems

Vertical Angles Theorem

Suitable Grade Level

Grades 8-10