Finite Math Examples

$8 x \cdot 1 + 9 x \cdot 2 = 117$ , $4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 1

Move variables to the left and constant terms to the right.

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Step 1.1

Simplify each term.

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Step 1.1.1

Multiply $8$ by $1$ .

$8 x + 9 x \cdot 2 = 117$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 1.1.2

Multiply $2$ by $9$ .

$8 x + 18 x = 117$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

$8 x + 18 x = 117$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 1.2

Add $8 x$ and $18 x$ .

$26 x = 117$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 1.3

Simplify each term.

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Step 1.3.1

Multiply $4$ by $1$ .

$26 x = 117$

$4 x + 6 x \cdot 2 = 66$

Step 1.3.2

Multiply $2$ by $6$ .

$26 x = 117$

$4 x + 12 x = 66$

$26 x = 117$

$4 x + 12 x = 66$

Step 1.4

Add $4 x$ and $12 x$ .

$26 x = 117$

$16 x = 66$

$26 x = 117$

$16 x = 66$

Step 2

Write the system as a matrix.

$[\begin{array}{c} 26 & 117 \\ 16 & 66 \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

Multiply each element of $R_{1}$ by $\frac{1}{26}$ to make the entry at $1, 1$ a $1$ .

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Step 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{26}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{26}{26} & \frac{117}{26} \\ 16 & 66 \end{array}]$

Step 3.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{9}{2} \\ 16 & 66 \end{array}]$

Step 3.2

Perform the row operation $R_{2} = R_{2} - 16 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 3.2.1

Perform the row operation $R_{2} = R_{2} - 16 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{2} \\ 16 - 16 \cdot 1 & 66 - 16 (\frac{9}{2}) \end{array}]$

Step 3.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{9}{2} \\ 0 & - 6 \end{array}]$

Step 3.3

Multiply each element of $R_{2}$ by $- \frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

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Step 3.3.1

Multiply each element of $R_{2}$ by $- \frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{9}{2} \\ - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot - 6 \end{array}]$

Step 3.3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{9}{2} \\ 0 & 1 \end{array}]$

Step 3.4

Perform the row operation $R_{1} = R_{1} - \frac{9}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 3.4.1

Perform the row operation $R_{1} = R_{1} - \frac{9}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{9}{2} \cdot 0 & \frac{9}{2} - \frac{9}{2} \cdot 1 \\ 0 & 1 \end{array}]$

Step 3.4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 4

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

$x = 0$

$0 = 1$

Step 5

The system is inconsistent so there is no solution.

$No solution$