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# Solving Linear Systems

## Solving Linear Systems 1.11 - Solution

a

The sum of a triangle's interior angles is $180 ^\circ.$ From the figure, we can see that one of the angles is a right angle. The remaining two are $x$ and $y.$ By adding these together, we can equate their sum with $180.$ $x+y+90=180$

b
Using the equation found in part A and the given equation, we can form a system of equations. $\begin{cases}x+y+90=180 & \, \text {(I)}\\ x-6=5y & \text {(II)}\end{cases}$ Let's solve it using the Substitution Method. To do so, we will start by isolating the $x\text{-}$variable in Equation (I). $x+y+90=180 \quad \Leftrightarrow \quad x=90-y$ Let's now substitute $90-y$ for $x$ in Equation (II) and solve the resulting equation for $y.$
$x-6=5y$
${\color{#0000FF}{90-y}}-6=5y$
Solve for $y$
$84-y=5y$
$84=6y$
$14=y$
$y=14$
To find the value of $x,$ we will substitute $14$ for $y$ in Equation (I).
$x+y+90=180$
$x+{\color{#0000FF}{14}}+90=180$
$x+104=180$
$x=76$
The solution to the system of equations is $(76,14).$ In the context of the problem, this means that the measures of the acute angles are $76^\circ$ and $14^\circ.$