Question

Use the given information to write and solve a system of linear equations to find the values of x and y. △STU≅△XYZ,m∠T=28∘,m∠U=(4x+y)∘,m∠X=130∘,m∠Y=(8x−6y)∘

Answer 1

To solve for x and y, set up two equations based on congruent angles in the triangles. Solve the system using substitution or elimination to get x = 25.25 and y = 29.

To solve the system of linear equations for the congruent triangles △STU and △XYZ, we need to set up two equations based on the information given. First, since the triangles are congruent, their corresponding angles are equal. This means that m∠U, which is given as (4x + y)°, must be equal to m∠X, which is 130°. The equation we can write from this is:

4x + y = 130 ... (1)

Similarly, m∠T in △STU is 28°, and because m∠T corresponds to m∠Z in △XYZ, and m∠Z is equal to m∠Y (since angles Z and Y are both opposite equal sides in the congruent triangles), the equation formed from the angles at Y and T is:

8x - 6y = 28 ... (2)

Now we have a system of equations to solve for x and y:

• 4x + y = 130
• 8x - 6y = 28

To solve this system, we can use the method of substitution or elimination. For simplicity, we can multiply equation (1) by 6 to eliminate y:

6(4x + y) = 6(130)

24x + 6y = 780 ... (3)

Now, by adding equation (2) and equation (3), we get:

24x + 6y + 8x - 6y = 780 + 28

32x = 808

x = 25.25

Substituting x back into equation (1) to solve for y, we get:

4(25.25) + y = 130

101 + y = 130

y = 29

Thus, the solution to the system of equations is x = 25.25 and y = 29.

Answer 2

Solving the system of linear equations derived, the values of x and y are calculated as: x = 5 and y = 2.

How to solve a system of equations?

Given that triangles STU and XYZ are congruent to each other, it means that:

m∠S = m∠X

m∠T = m∠Y, and

m∠U = m∠Z [m∠Z = 180 - 130 - 28 = 22°]

Therefore, we can find the values of x and y by creating a system of equations which can be formed as shown:

4x + y = 22 [eqn 1]

8x - 6y = 28 [eqn 2]

To eliminate one variable, let's multiply the first equation by 6 (to make the coefficients of y in both equations equal):

6(4x + y) = 6 * 22

24x + 6y = 132 (eqn 3)

Now, we can add equation (3) to equation (2) to eliminate y:

(24x + 6y) + (8x - 6y) = 132 + 28

32x = 160

Divide both sides by 32 to solve for x:

x = 5

Now that we have x, we can substitute it back into one of the original equations, let's use equation (1):

4(5) + y = 22

20 + y = 22

Subtract 20 from both sides:

y = 2

So, the solution to the system of equations is x = 5 and y = 2.