Question

mathematical connections solve a system of linear equations to find the values of x and y. the linear equations are 4x + □ = 180 and 2x + □ = 180. the solutions are x = □ and y = □

Explanation:

Step1: Determine the equations

In a trapezoid, consecutive - angles between the bases are supplementary.
For the pair of angles \(4x^{\circ}\) and \((2x + 12)^{\circ}\), we have the equation \(4x+(2x + 12)=180\) (because they are same - side interior angles).
For the pair of angles \(2y^{\circ}\) and \((y + 6)^{\circ}\), we have the equation \(2y+(y + 6)=180\) (because they are same - side interior angles).

Step2: Solve the first equation \(4x+(2x + 12)=180\)

Combine like terms:

$$
LATEXBLOCK0
$$

Subtract 12 from both sides: \(6x=180 - 12=168\).
Divide both sides by 6: \(x=\frac{168}{6}=28\).

Step3: Solve the second equation \(2y+(y + 6)=180\)

Combine like terms:

$$
LATEXBLOCK1
$$

Subtract 6 from both sides: \(3y=180 - 6 = 174\).
Divide both sides by 3: \(y=\frac{174}{3}=58\).

Answer:

The linear equations are \(4x+(2x + 12)=180\) and \(2y+(y + 6)=180\). The solutions are \(x = 28\) and \(y = 58\).