Question

measurement and geometry continued

1. determine the values of angles \\(x\\) and \\(y\\).

algebra - solving equations
solve each of the following equations. leave answers as fractions, reduced to lowest terms, where applicable.
a) \\(3x - 5 = 22\\)
b) \\(2(x + 3) = 5x + 9\\)
c) \\(\frac{x - 1}{2} = \frac{x - 5}{3}\\)

Explanation:

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<reasoning>

Calculate regular pentagon interior and exterior angles

\[

$$\begin{aligned} y &= \frac{(5-2) \times 180^\circ}{5} = 108^\circ \\ x &= 180^\circ - y = 180^\circ - 108^\circ = 72^\circ \end{aligned}$$

\]

Solve equation a

\[

$$\begin{aligned} 3x - 5 &= 22 \\ 3x &= 27 \\ x &= 9 \end{aligned}$$

\]

Solve equation b

\[

$$\begin{aligned} 2(x + 3) &= 5x + 9 \\ 2x + 6 &= 5x + 9 \\ -3 &= 3x \\ x &= -1 \end{aligned}$$

\]

Solve equation c

\[

$$\begin{aligned} \frac{x-1}{2} &= \frac{x-5}{8} \\ 8(x - 1) &= 2(x - 5) \\ 8x - 8 &= 2x - 10 \\ 6x &= -2 \\ x &= -\frac{1}{3} \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

\(x = 72^\circ\), \(y = 108^\circ\)

Question 2

\(x = 9\)

Question 3

\(x = -1\)

Question 4

\(x = -\frac{1}{3}\)
</answer>

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Answer:

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<reasoning>

Calculate regular pentagon interior and exterior angles

\[

$$\begin{aligned} y &= \frac{(5-2) \times 180^\circ}{5} = 108^\circ \\ x &= 180^\circ - y = 180^\circ - 108^\circ = 72^\circ \end{aligned}$$

\]

Solve equation a

\[

$$\begin{aligned} 3x - 5 &= 22 \\ 3x &= 27 \\ x &= 9 \end{aligned}$$

\]

Solve equation b

\[

$$\begin{aligned} 2(x + 3) &= 5x + 9 \\ 2x + 6 &= 5x + 9 \\ -3 &= 3x \\ x &= -1 \end{aligned}$$

\]

Solve equation c

\[

$$\begin{aligned} \frac{x-1}{2} &= \frac{x-5}{8} \\ 8(x - 1) &= 2(x - 5) \\ 8x - 8 &= 2x - 10 \\ 6x &= -2 \\ x &= -\frac{1}{3} \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

\(x = 72^\circ\), \(y = 108^\circ\)

Question 2

\(x = 9\)

Question 3

\(x = -1\)

Question 4

\(x = -\frac{1}{3}\)
</answer>

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"Mathematics",
"Algebra",
"Linear Equations"
]
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