Finite Math Examples

$8 x + 7 y = - 34$ , $16 + 14 y = - 36$

Step 1

Move all terms not containing a variable to the right side of the equation.

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Step 1.1

Subtract $16$ from both sides of the equation.

$8 x + 7 y = - 34$

$14 y = - 36 - 16$

Step 1.2

Subtract $16$ from $- 36$ .

$8 x + 7 y = - 34$

$14 y = - 52$

$8 x + 7 y = - 34$

$14 y = - 52$

Step 2

Write the system as a matrix.

$[\begin{array}{c} 8 & 7 & - 34 \\ 0 & 14 & - 52 \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

Multiply each element of $R_{1}$ by $\frac{1}{8}$ to make the entry at $1, 1$ a $1$ .

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Step 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{8}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{8}{8} & \frac{7}{8} & \frac{- 34}{8} \\ 0 & 14 & - 52 \end{array}]$

Step 3.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{7}{8} & - \frac{17}{4} \\ 0 & 14 & - 52 \end{array}]$

Step 3.2

Multiply each element of $R_{2}$ by $\frac{1}{14}$ to make the entry at $2, 2$ a $1$ .

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Step 3.2.1

Multiply each element of $R_{2}$ by $\frac{1}{14}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{7}{8} & - \frac{17}{4} \\ \frac{0}{14} & \frac{14}{14} & \frac{- 52}{14} \end{array}]$

Step 3.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{7}{8} & - \frac{17}{4} \\ 0 & 1 & - \frac{26}{7} \end{array}]$

Step 3.3

Perform the row operation $R_{1} = R_{1} - \frac{7}{8} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 3.3.1

Perform the row operation $R_{1} = R_{1} - \frac{7}{8} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{7}{8} \cdot 0 & \frac{7}{8} - \frac{7}{8} \cdot 1 & - \frac{17}{4} - \frac{7}{8} (- \frac{26}{7}) \\ 0 & 1 & - \frac{26}{7} \end{array}]$

Step 3.3.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 1 \\ 0 & 1 & - \frac{26}{7} \end{array}]$

Step 4

Use the result matrix to declare the final solution to the system of equations.

$x = - 1$

$y = - \frac{26}{7}$

Step 5

The solution is the set of ordered pairs that make the system true.

$(- 1, - \frac{26}{7})$