Finite Math Examples

$x \cdot 1 + 2 x \cdot 2 + 8 x \cdot 3 = 2$ , $x \cdot 1 + x \cdot 2 + 4 x \cdot 3 = 1$

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 $1$ .

$x + 2 x \cdot 2 + 8 x \cdot 3 = 2$

$x \cdot 1 + x \cdot 2 + 4 x \cdot 3 = 1$

Step 1.1.2

Multiply $2$ by $2$ .

$x + 4 x + 8 x \cdot 3 = 2$

$x \cdot 1 + x \cdot 2 + 4 x \cdot 3 = 1$

Step 1.1.3

Multiply $3$ by $8$ .

$x + 4 x + 24 x = 2$

$x \cdot 1 + x \cdot 2 + 4 x \cdot 3 = 1$

$x + 4 x + 24 x = 2$

$x \cdot 1 + x \cdot 2 + 4 x \cdot 3 = 1$

Step 1.2

Add $x$ and $4 x$ .

$5 x + 24 x = 2$

$x \cdot 1 + x \cdot 2 + 4 x \cdot 3 = 1$

Step 1.3

Add $5 x$ and $24 x$ .

$29 x = 2$

$x \cdot 1 + x \cdot 2 + 4 x \cdot 3 = 1$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $x$ by $1$ .

$29 x = 2$

$x + x \cdot 2 + 4 x \cdot 3 = 1$

Step 1.4.2

Move $2$ to the left of $x$ .

$29 x = 2$

$x + 2 \cdot x + 4 x \cdot 3 = 1$

Step 1.4.3

Multiply $3$ by $4$ .

$29 x = 2$

$x + 2 x + 12 x = 1$

$29 x = 2$

$x + 2 x + 12 x = 1$

Step 1.5

Add $x$ and $2 x$ .

$29 x = 2$

$3 x + 12 x = 1$

Step 1.6

Add $3 x$ and $12 x$ .

$29 x = 2$

$15 x = 1$

$29 x = 2$

$15 x = 1$

Step 2

Write the system as a matrix.

$[\begin{array}{c} 29 & 2 \\ 15 & 1 \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

Multiply each element of $R_{1}$ by $\frac{1}{29}$ to make the entry at $1, 1$ a $1$ .

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Step 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{29}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{29}{29} & \frac{2}{29} \\ 15 & 1 \end{array}]$

Step 3.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{29} \\ 15 & 1 \end{array}]$

$[\begin{array}{c} 1 & \frac{2}{29} \\ 15 & 1 \end{array}]$

Step 3.2

Perform the row operation $R_{2} = R_{2} - 15 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 3.2.1

Perform the row operation $R_{2} = R_{2} - 15 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{29} \\ 15 - 15 \cdot 1 & 1 - 15 (\frac{2}{29}) \end{array}]$

Step 3.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{29} \\ 0 & - \frac{1}{29} \end{array}]$

$[\begin{array}{c} 1 & \frac{2}{29} \\ 0 & - \frac{1}{29} \end{array}]$

Step 3.3

Multiply each element of $R_{2}$ by $- 29$ to make the entry at $2, 2$ a $1$ .

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Step 3.3.1

Multiply each element of $R_{2}$ by $- 29$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{29} \\ - 29 \cdot 0 & - 29 (- \frac{1}{29}) \end{array}]$

Step 3.3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{29} \\ 0 & 1 \end{array}]$

$[\begin{array}{c} 1 & \frac{2}{29} \\ 0 & 1 \end{array}]$

Step 3.4

Perform the row operation $R_{1} = R_{1} - \frac{2}{29} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 3.4.1

Perform the row operation $R_{1} = R_{1} - \frac{2}{29} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{2}{29} \cdot 0 & \frac{2}{29} - \frac{2}{29} \cdot 1 \\ 0 & 1 \end{array}]$

Step 3.4.2

Simplify $R_{1}$ .

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

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

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

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.

$No solution$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$