Finite Math Examples

$x \cdot 1 + 2 x \cdot 2 = 21$ , $- 3 x \cdot 1 - 7 x \cdot 2 = - 71$

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 $x$ by $1$ .

$x + 2 x \cdot 2 = 21$

$- 3 x \cdot 1 - 7 x \cdot 2 = - 71$

Step 1.1.2

Multiply $2$ by $2$ .

$x + 4 x = 21$

$- 3 x \cdot 1 - 7 x \cdot 2 = - 71$

$x + 4 x = 21$

$- 3 x \cdot 1 - 7 x \cdot 2 = - 71$

Step 1.2

Add $x$ and $4 x$ .

$5 x = 21$

$- 3 x \cdot 1 - 7 x \cdot 2 = - 71$

Step 1.3

Simplify each term.

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Step 1.3.1

Multiply $- 3$ by $1$ .

$5 x = 21$

$- 3 x - 7 x \cdot 2 = - 71$

Step 1.3.2

Multiply $2$ by $- 7$ .

$5 x = 21$

$- 3 x - 14 x = - 71$

$5 x = 21$

$- 3 x - 14 x = - 71$

Step 1.4

Subtract $14 x$ from $- 3 x$ .

$5 x = 21$

$- 17 x = - 71$

$5 x = 21$

$- 17 x = - 71$

Step 2

Write the system as a matrix.

$[\begin{array}{c} 5 & 21 \\ - 17 & - 71 \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

Multiply each element of $R_{1}$ by $\frac{1}{5}$ to make the entry at $1, 1$ a $1$ .

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Step 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{5}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{5}{5} & \frac{21}{5} \\ - 17 & - 71 \end{array}]$

Step 3.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{21}{5} \\ - 17 & - 71 \end{array}]$

Step 3.2

Perform the row operation $R_{2} = R_{2} + 17 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 3.2.1

Perform the row operation $R_{2} = R_{2} + 17 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{21}{5} \\ - 17 + 17 \cdot 1 & - 71 + 17 (\frac{21}{5}) \end{array}]$

Step 3.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{21}{5} \\ 0 & \frac{2}{5} \end{array}]$

Step 3.3

Multiply each element of $R_{2}$ by $\frac{5}{2}$ to make the entry at $2, 2$ a $1$ .

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Step 3.3.1

Multiply each element of $R_{2}$ by $\frac{5}{2}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{21}{5} \\ \frac{5}{2} \cdot 0 & \frac{5}{2} \cdot \frac{2}{5} \end{array}]$

Step 3.3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{21}{5} \\ 0 & 1 \end{array}]$

Step 3.4

Perform the row operation $R_{1} = R_{1} - \frac{21}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 3.4.1

Perform the row operation $R_{1} = R_{1} - \frac{21}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{21}{5} \cdot 0 & \frac{21}{5} - \frac{21}{5} \cdot 1 \\ 0 & 1 \end{array}]$

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.

$No solution$