Question

ming drew a triangle. the sum of the angles of the triangle is 180°. in mings triangle, ∠b is twice the measure of ∠a, and angle ∠c is 5° greater than four times ∠a. the system of equations below can be used to find the measure of the three angles, where a represents the measure of ∠a, b represents the measure of ∠b, and c represents the measure of ∠c

$a + b + c = 180$
$2a = b$
$4a + 5 = c$

which matrix represents the measure of each angle in mings triangle?

$\

$$\begin{bmatrix}1&1&0&75\\\\0&1&0&50\\\\0&0&1&105\\end{bmatrix}$$

$

$\

$$\begin{bmatrix}1&1&1&180\\end{bmatrix}$$

$

Explanation:

Step1: Substitute \(b = 2a\) and \(c=4a + 5\) into \(a + b + c=180\)

Substitute the expressions for \(b\) and \(c\) into the first - equation:
\(a+(2a)+(4a + 5)=180\)
Combine like terms:
\(a+2a+4a+5 = 180\)
\(7a+5 = 180\)

Step2: Solve for \(a\)

Subtract 5 from both sides of the equation \(7a+5 = 180\):
\(7a=180 - 5\)
\(7a=175\)
Divide both sides by 7:
\(a=\frac{175}{7}=25\)

Step3: Solve for \(b\)

Since \(b = 2a\), substitute \(a = 25\) into the equation:
\(b=2\times25 = 50\)

Step4: Solve for \(c\)

Since \(c=4a + 5\), substitute \(a = 25\) into the equation:
\(c=4\times25+5\)
\(c = 100 + 5=105\)

Answer:

The matrix \(

$$\begin{bmatrix}1&1&1&|&180\\1&-2&0&|&0\\4&0&-1&|& - 5\end{bmatrix}$$

\) can be used to solve the system \(a + b + c=180\), \(b = 2a\) (or \(a-2b=0\)), \(c=4a + 5\) (or \(4a - c=-5\)). After solving \(a = 25\), \(b = 50\), \(c = 105\), the matrix representing the measures of the angles is \(

$$\begin{bmatrix}1&0&0&|&25\\0&1&0&|&50\\0&0&1&|&105\end{bmatrix}$$

\)