Finite Math Examples

$p [\begin{array}{c} 6 & 7 \\ 8 & 9 \end{array}] = [\begin{array}{c} 2 & 3 \\ 4 & 5 \end{array}]$

Step 1

Multiply $p$ by each element of the matrix.

$[\begin{array}{c} p \cdot 6 & p \cdot 7 \\ p \cdot 8 & p \cdot 9 \end{array}] = [\begin{array}{c} 2 & 3 \\ 4 & 5 \end{array}]$

Step 2

Simplify each element in the matrix.

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Step 2.1

Move $6$ to the left of $p$ .

$[\begin{array}{c} 6 p & p \cdot 7 \\ p \cdot 8 & p \cdot 9 \end{array}] = [\begin{array}{c} 2 & 3 \\ 4 & 5 \end{array}]$

Step 2.2

Move $7$ to the left of $p$ .

$[\begin{array}{c} 6 p & 7 p \\ p \cdot 8 & p \cdot 9 \end{array}] = [\begin{array}{c} 2 & 3 \\ 4 & 5 \end{array}]$

Step 2.3

Move $8$ to the left of $p$ .

$[\begin{array}{c} 6 p & 7 p \\ 8 p & p \cdot 9 \end{array}] = [\begin{array}{c} 2 & 3 \\ 4 & 5 \end{array}]$

Step 2.4

Move $9$ to the left of $p$ .

$[\begin{array}{c} 6 p & 7 p \\ 8 p & 9 p \end{array}] = [\begin{array}{c} 2 & 3 \\ 4 & 5 \end{array}]$

Step 3

Write as a linear system of equations.

$6 p = 2$

$7 p = 3$

$8 p = 4$

$9 p = 5$

Step 4

Solve the system of equations.

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Step 4.1

Divide each term in $6 p = 2$ by $6$ and simplify.

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Step 4.1.1

Divide each term in $6 p = 2$ by $6$ .

$\frac{6 p}{6} = \frac{2}{6}$

$7 p = 3$

$8 p = 4$

$9 p = 5$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $6$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{6 p}{6} = \frac{2}{6}$

$7 p = 3$

$8 p = 4$

$9 p = 5$

Step 4.1.2.1.2

Divide $p$ by $1$ .

$p = \frac{2}{6}$

$7 p = 3$

$8 p = 4$

$9 p = 5$

$p = \frac{2}{6}$

$7 p = 3$

$8 p = 4$

$9 p = 5$

$p = \frac{2}{6}$

$7 p = 3$

$8 p = 4$

$9 p = 5$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Cancel the common factor of $2$ and $6$ .

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Step 4.1.3.1.1

Factor $2$ out of $2$ .

$p = \frac{2 (1)}{6}$

$7 p = 3$

$8 p = 4$

$9 p = 5$

Step 4.1.3.1.2

Cancel the common factors.

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Step 4.1.3.1.2.1

Factor $2$ out of $6$ .

$p = \frac{2 \cdot 1}{2 \cdot 3}$

$7 p = 3$

$8 p = 4$

$9 p = 5$

Step 4.1.3.1.2.2

Cancel the common factor.

$p = \frac{2 \cdot 1}{2 \cdot 3}$

$7 p = 3$

$8 p = 4$

$9 p = 5$

Step 4.1.3.1.2.3

Rewrite the expression.

$p = \frac{1}{3}$

$7 p = 3$

$8 p = 4$

$9 p = 5$

$p = \frac{1}{3}$

$7 p = 3$

$8 p = 4$

$9 p = 5$

$p = \frac{1}{3}$

$7 p = 3$

$8 p = 4$

$9 p = 5$

$p = \frac{1}{3}$

$7 p = 3$

$8 p = 4$

$9 p = 5$

$p = \frac{1}{3}$

$7 p = 3$

$8 p = 4$

$9 p = 5$

Step 4.2

Replace all occurrences of $p$ with $\frac{1}{3}$ in each equation.

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Step 4.2.1

Replace all occurrences of $p$ in $7 p = 3$ with $\frac{1}{3}$ .

$7 (\frac{1}{3}) = 3$

$p = \frac{1}{3}$

$8 p = 4$

$9 p = 5$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Combine $7$ and $\frac{1}{3}$ .

$\frac{7}{3} = 3$

$p = \frac{1}{3}$

$8 p = 4$

$9 p = 5$

$\frac{7}{3} = 3$

$p = \frac{1}{3}$

$8 p = 4$

$9 p = 5$

Step 4.2.3

Replace all occurrences of $p$ in $8 p = 4$ with $\frac{1}{3}$ .

$8 (\frac{1}{3}) = 4$

$\frac{7}{3} = 3$

$p = \frac{1}{3}$

$9 p = 5$

Step 4.2.4

Simplify the left side.

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Step 4.2.4.1

Combine $8$ and $\frac{1}{3}$ .

$\frac{8}{3} = 4$

$\frac{7}{3} = 3$

$p = \frac{1}{3}$

$9 p = 5$

$\frac{8}{3} = 4$

$\frac{7}{3} = 3$

$p = \frac{1}{3}$

$9 p = 5$

Step 4.2.5

Replace all occurrences of $p$ in $9 p = 5$ with $\frac{1}{3}$ .

$9 (\frac{1}{3}) = 5$

$\frac{8}{3} = 4$

$\frac{7}{3} = 3$

$p = \frac{1}{3}$

Step 4.2.6

Simplify the left side.

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Step 4.2.6.1

Cancel the common factor of $3$ .

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Step 4.2.6.1.1

Factor $3$ out of $9$ .

$3 (3) (\frac{1}{3}) = 5$

$\frac{8}{3} = 4$

$\frac{7}{3} = 3$

$p = \frac{1}{3}$

Step 4.2.6.1.2

Cancel the common factor.

$3 \cdot (3 (\frac{1}{3})) = 5$

$\frac{8}{3} = 4$

$\frac{7}{3} = 3$

$p = \frac{1}{3}$

Step 4.2.6.1.3

Rewrite the expression.

$3 = 5$

$\frac{8}{3} = 4$

$\frac{7}{3} = 3$

$p = \frac{1}{3}$

$3 = 5$

$\frac{8}{3} = 4$

$\frac{7}{3} = 3$

$p = \frac{1}{3}$

$3 = 5$

$\frac{8}{3} = 4$

$\frac{7}{3} = 3$

$p = \frac{1}{3}$

$3 = 5$

$\frac{8}{3} = 4$

$\frac{7}{3} = 3$

$p = \frac{1}{3}$

Step 4.3

Since $3 = 5$ is not true, there is no solution.

$No solution$