Solving Linear Systems

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

Substitute

x={\color{#0000FF}{90-y}}

{\color{#0000FF}{90-y}}-6=5y

Solve for

y

SubTermSubtract term

84-y=5y

AddEqn

\text{LHS}+y=\text{RHS}+y

84=6y

DivEqn

\left.\text{LHS}\middle/6\right.=\left.\text{RHS}\middle/6\right.

14=y

RearrangeEqnRearrange equation

y=14

To find the value of

x,

we will substitute

14

for

y

in Equation (I).

x+y+90=180

Substitute

y={\color{#0000FF}{14}}

x+{\color{#0000FF}{14}}+90=180

AddTermsAdd terms

x+104=180

SubEqn

\text{LHS}-104=\text{RHS}-104

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.

Solving Linear Systems 1.11 - Solution