Solve each system of equations 8x−6z=382x−5y+3z=5x+10y−4z=8

Q15. - Algebra 2 | StudyQuestionHub

Q15.

Question

Solve each system of equations

$\begin{array}{l} 8 x - 6 z = 38 \\ 2 x - 5 y + 3 z = 5 \\ x + 10 y - 4 z = 8 \end{array}$

Step-by-Step Solution

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Answer

The solution set of the given system of equations is $(4, 0, - 1)$ .

1Step 1 – Use the elimination method to get the system of equations in two variables.

Multiply the equation $2 x - 5 y + 3 z = 5$ by $2$ and add the new resultant equation to $x + 10 y - 4 z = 8$ .

$\begin{array}{l} \underline{\begin{array}{l} 2 x - 5 y + 3 z = 5 \\ x + 10 y - 4 z = 8 \end{array}} \begin{matrix} multiply by 2 \end{matrix} \underline{\begin{array}{l} 4 x - 10 y + 6 z = 10 \\ x + 10 y - 4 z = 8 \end{array}} \\ 5 x + 0 + 2 z = 18 \end{array}$

So, the resultant equation is $5 x + 2 z = 18$ .

2Step 2 – Use the elimination method to solve the system of two equations.

Multiply $5 x + 2 z = 18$ by 3 and add the new resultant equation to $8 x - 6 z = 38$ .

$\begin{array}{l} \underline{\begin{array}{l} 5 x + 2 z = 18 \\ 8 x - 6 z = 38 \end{array}} \begin{matrix} multiply by 3 \end{matrix} \underline{\begin{array}{l} 15 x + 6 z = 54 \\ 8 x - 6 z = 38 \end{array}} \\ 23 x + 0 = 92 \end{array}$

Solve for : $\begin{array}{l} 23 x = 92 \\ \frac{23 x}{23} = \frac{92}{23} Divide both sides by 23 \\ x = 4 \end{array}$

3Step 3 – Find the values of y and z .

Substitute $x = 4 i n 5 x + 2 z = 18$ and find the value of $z$ .

$\begin{array}{l} 5 x + 2 z = 18 \\ 5 (4) + 2 z = 18 Substitute 4 for x \\ 20 + 2 z = 18 Simplify \\ 2 z = - 2 Subtract 20 from both sides \\ z = - 1 Divide both sides by 2 \end{array}$

Substitute $x = 4, z = - 1 i n 2 x - 5 y + 3 z = 5$ and find the value of $y$

$\begin{array}{l} 2 x - 5 y + 3 z = 5 \\ 2 (4) - 5 y + 3 (- 1) = 5 substitute 4 for x, - 1 for z \\ 8 - 5 y - 3 = 5 simplify \\ 5 - 5 y = 5 simplify \\ - 5 y = 0 subtract 5 from both sides \\ y = 0 divide both sides by - 5 \end{array}$

Hence, the solution of the given system of equations is $(x, y, z) = (4, 0, - 1)$ .

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