Question 146410
let x=measure of angle 1 and y=measure of angle 2


"Two angles are complementary" tells us that the first equation is {{{x+y=90}}}



"The sum of the measure of the first angle and one-fourth the second angle is 63.75 degrees" translates to this equation:

{{{x+(1/4)y=63.75}}}



{{{4(x+(1/4)y)=4(63.75)}}} Multiply both sides by the LCD 4 to clear the fraction



{{{4x+y=255}}} Multiply



So we have the system of equations:



{{{system(x+y=90,4x+y=255)}}}



Now let's use substitution to solve this system





Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.





So let's isolate y in the first equation


{{{x+y=90}}} Start with the first equation



{{{y=90-x}}}  Subtract {{{x}}} from both sides



{{{y=-x+90}}} Rearrange the equation





---------------------


Since {{{y=-x+90}}}, we can now replace each {{{y}}} in the second equation with {{{-x+90}}} to solve for {{{x}}}




{{{4x+highlight((-x+90))=255}}} Plug in {{{y=-x+90}}} into the first equation. In other words, replace each {{{y}}} with {{{-x+90}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{3x+90=255}}} Combine like terms on the left side



{{{3x=255-90}}}Subtract 90 from both sides



{{{3x=165}}} Combine like terms on the right side



{{{x=(165)/(3)}}} Divide both sides by 3 to isolate x




{{{x=55}}} Divide






-----------------First Answer------------------------------



So the first part of our answer is: {{{x=55}}}










Since we know that {{{x=55}}} we can plug it into the equation {{{y=-x+90}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=-x+90}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=-(55)+90}}} Plug in {{{x=55}}}



{{{y=-55+90}}} Multiply



{{{y=35}}} Combine like terms 




-----------------Second Answer------------------------------



So the second part of our answer is: {{{y=35}}}










-----------------Summary------------------------------


So our answers are:


{{{x=55}}} and {{{y=35}}}



So the measure of smaller angle is 35 degrees and the measure of the other angle is 55 degrees