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Solving Linear Systems

Solving Linear Systems 1.11 - Solution

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a

The sum of a triangle's interior angles is 180.180 ^\circ. From the figure, we can see that one of the angles is a right angle. The remaining two are xx and y.y. By adding these together, we can equate their sum with 180.180. x+y+90=180 x+y+90=180

b
Using the equation found in part A and the given equation, we can form a system of equations. {x+y+90=180(I)x6=5y(II) \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-x\text{-}variable in Equation (I). x+y+90=180x=90y x+y+90=180 \quad \Leftrightarrow \quad x=90-y Let's now substitute 90y90-y for xx in Equation (II) and solve the resulting equation for y.y.
x6=5yx-6=5y
90y6=5y{\color{#0000FF}{90-y}}-6=5y
Solve for yy
84y=5y84-y=5y
84=6y84=6y
14=y14=y
y=14y=14
To find the value of x,x, we will substitute 1414 for yy in Equation (I).
x+y+90=180x+y+90=180
x+14+90=180x+{\color{#0000FF}{14}}+90=180
x+104=180x+104=180
x=76x=76
The solution to the system of equations is (76,14).(76,14). In the context of the problem, this means that the measures of the acute angles are 7676^\circ and 14.14^\circ.