Question

Translate to a system of equations and then solve:

Two angles are supplementary. The measure of the larger angle is $18$less than twice the measure of the smaller angle. Find the measures of the angles.

Short answer

The measure of the angles are ${66}^{\circ }$and ${114}^{\circ }$.

Step-by-step solution

Step 1. Given

We are given two supplementary angles and larger angle is $18$less than twice the measure of 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

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

Also, the other equation will be $y=2x-18$.

Step 3. Solving the equations

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

localid="1644389081295" $$\begin{gathered} x+2x-18={180}^{\circ } \\ 3x-18={180}^{\circ } \end{gathered}$$

Adding $18$both sides, we get

localid="1644389531085" $3x={198}^{\circ }$

Dividing by $3$,

$x={66}^{\circ }$

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

$$\begin{gathered} y=2\times 66-18 \\ ={114}^{\circ } \end{gathered}$$

Step 4. Checking the solution  

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

$$\begin{gathered} x+y={180}^{\circ } \\ {66}^{\circ }+{114}^{\circ }={180}^{\circ } \\ {180}^{\circ }={180}^{\circ } \\ y=2x-14 \\ {114}^{\circ }=2\left({66}^{\circ }\right)-18 \\ {114}^{\circ }={132}^{\circ }-18 \\ {114}^{\circ }={114}^{\circ } \end{gathered}$$

This is true, hence the solution is correct.