Question

Translate to a system of equations and then solve:

Two angles are supplementary. The measure of the larger angle is $12$degrees more than three times the smaller angle. Find the measures of the angles.

Short answer

The measure of the angles are ${42}^{\circ }$and ${138}^{\circ }$.

Step-by-step solution

Step 1. Given

We are given two supplementary angles and larger angle is $12$degrees more than three times the smaller angle.

Step 2. Assumptions and formation of equations.

Let the smaller angle be $x$and larger angle be $y$,

We know that the sum of supplementary angles is equal to ${180}^{\circ }$. so

$x+y={180}^{\circ }$

And the other equation will be,

role="math" localid="1644640052444" $y=3x+12$

Step 3. Solving the equations

Putting the value of $y$in first equation, we get

role="math" localid="1644640100197" $$\begin{gathered} x+y={180}^{\circ } \\ x+3x+12={180}^{\circ } \\ 4x={168}^{\circ } \\ x=\frac{168}{4} \\ x={42}^{\circ } \end{gathered}$$

Putting the value of $x$in second equation, we get

role="math" localid="1644640169512" $$\begin{gathered} y=3x+12 \\ y=3\times 42+12 \\ y=126+12 \\ y={138}^{\mathbf{\circ }} \end{gathered}$$

Step 4. Checking the solution

Checking the solution by putting the value of $x,y$in the equations, we get

$$\begin{gathered} x+y={180}^{\circ } \\ {42}^{\circ }+{138}^{\circ }={180}^{\circ } \\ {180}^{\circ }={180}^{\circ } \\ y=3x+12 \\ {138}^{\circ }=3\left({42}^{\circ }\right)+12 \\ {138}^{\circ }={126}^{\circ }+12 \\ {138}^{\circ }={138}^{\circ } \end{gathered}$$

This is true, hence the solution is correct.