Algebra Examples

$[\begin{array}{c} 6 & 5 \\ 5 & 4 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 2 \\ 1 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 6 & 5 \\ 5 & 4 \end{array}] [\begin{array}{c} x \\ y \end{array}]$ .

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Step 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $2 \times 2$ and the second matrix is $2 \times 1$ .

Step 1.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 6 x + 5 y \\ 5 x + 4 y \end{array}] = [\begin{array}{c} 2 \\ 1 \end{array}]$

Step 2

The matrix equation can be written as a set of equations.

$6 x + 5 y = 2$

$5 x + 4 y = 1$

Step 3

Solve for $x$ in $6 x + 5 y = 2$ .

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Step 3.1

Subtract $5 y$ from both sides of the equation.

$6 x = 2 - 5 y$

$5 x + 4 y = 1$

Step 3.2

Divide each term in $6 x = 2 - 5 y$ by $6$ and simplify.

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Step 3.2.1

Divide each term in $6 x = 2 - 5 y$ by $6$ .

$\frac{6 x}{6} = \frac{2}{6} + \frac{- 5 y}{6}$

$5 x + 4 y = 1$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $6$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{2}{6} + \frac{- 5 y}{6}$

$5 x + 4 y = 1$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{6} + \frac{- 5 y}{6}$

$5 x + 4 y = 1$

$x = \frac{2}{6} + \frac{- 5 y}{6}$

$5 x + 4 y = 1$

$x = \frac{2}{6} + \frac{- 5 y}{6}$

$5 x + 4 y = 1$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

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

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Step 3.2.3.1.1.1

Factor $2$ out of $2$ .

$x = \frac{2 (1)}{6} + \frac{- 5 y}{6}$

$5 x + 4 y = 1$

Step 3.2.3.1.1.2

Cancel the common factors.

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Step 3.2.3.1.1.2.1

Factor $2$ out of $6$ .

$x = \frac{2 \cdot 1}{2 \cdot 3} + \frac{- 5 y}{6}$

$5 x + 4 y = 1$

Step 3.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 1}{2 \cdot 3} + \frac{- 5 y}{6}$

$5 x + 4 y = 1$

Step 3.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{1}{3} + \frac{- 5 y}{6}$

$5 x + 4 y = 1$

$x = \frac{1}{3} + \frac{- 5 y}{6}$

$5 x + 4 y = 1$

$x = \frac{1}{3} + \frac{- 5 y}{6}$

$5 x + 4 y = 1$

Step 3.2.3.1.2

Move the negative in front of the fraction.

$x = \frac{1}{3} - \frac{5 y}{6}$

$5 x + 4 y = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

$5 x + 4 y = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

$5 x + 4 y = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

$5 x + 4 y = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

$5 x + 4 y = 1$

Step 4

Replace all occurrences of $x$ with $\frac{1}{3} - \frac{5 y}{6}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $5 x + 4 y = 1$ with $\frac{1}{3} - \frac{5 y}{6}$ .

$5 (\frac{1}{3} - \frac{5 y}{6}) + 4 y = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $5 (\frac{1}{3} - \frac{5 y}{6}) + 4 y$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$5 (\frac{1}{3}) + 5 (- \frac{5 y}{6}) + 4 y = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.1.2

Combine $5$ and $\frac{1}{3}$ .

$\frac{5}{3} + 5 (- \frac{5 y}{6}) + 4 y = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.1.3

Multiply $5 (- \frac{5 y}{6})$ .

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Step 4.2.1.1.3.1

Multiply $- 1$ by $5$ .

$\frac{5}{3} - 5 \frac{5 y}{6} + 4 y = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.1.3.2

Combine $- 5$ and $\frac{5 y}{6}$ .

$\frac{5}{3} + \frac{- 5 (5 y)}{6} + 4 y = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.1.3.3

Multiply $5$ by $- 5$ .

$\frac{5}{3} + \frac{- 25 y}{6} + 4 y = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

$\frac{5}{3} + \frac{- 25 y}{6} + 4 y = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.1.4

Move the negative in front of the fraction.

$\frac{5}{3} - \frac{25 y}{6} + 4 y = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

$\frac{5}{3} - \frac{25 y}{6} + 4 y = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.2

To write $4 y$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{5}{3} - \frac{25 y}{6} + 4 y \cdot \frac{6}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.3

Simplify terms.

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Step 4.2.1.3.1

Combine $4 y$ and $\frac{6}{6}$ .

$\frac{5}{3} - \frac{25 y}{6} + \frac{4 y \cdot 6}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.3.2

Combine the numerators over the common denominator.

$\frac{5}{3} + \frac{- 25 y + 4 y \cdot 6}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

$\frac{5}{3} + \frac{- 25 y + 4 y \cdot 6}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.4

Simplify each term.

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Step 4.2.1.4.1

Simplify the numerator.

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Step 4.2.1.4.1.1

Factor $y$ out of $- 25 y + 4 y \cdot 6$ .

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Step 4.2.1.4.1.1.1

Factor $y$ out of $- 25 y$ .

$\frac{5}{3} + \frac{y \cdot - 25 + 4 y \cdot 6}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.4.1.1.2

Factor $y$ out of $4 y \cdot 6$ .

$\frac{5}{3} + \frac{y \cdot - 25 + y (4 \cdot 6)}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.4.1.1.3

Factor $y$ out of $y \cdot - 25 + y (4 \cdot 6)$ .

$\frac{5}{3} + \frac{y (- 25 + 4 \cdot 6)}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

$\frac{5}{3} + \frac{y (- 25 + 4 \cdot 6)}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.4.1.2

Multiply $4$ by $6$ .

$\frac{5}{3} + \frac{y (- 25 + 24)}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.4.1.3

Add $- 25$ and $24$ .

$\frac{5}{3} + \frac{y \cdot - 1}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

$\frac{5}{3} + \frac{y \cdot - 1}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.4.2

Move $- 1$ to the left of $y$ .

$\frac{5}{3} + \frac{- 1 \cdot y}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 4.2.1.4.3

Move the negative in front of the fraction.

$\frac{5}{3} - \frac{y}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

$\frac{5}{3} - \frac{y}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

$\frac{5}{3} - \frac{y}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

$\frac{5}{3} - \frac{y}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

$\frac{5}{3} - \frac{y}{6} = 1$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5

Solve for $y$ in $\frac{5}{3} - \frac{y}{6} = 1$ .

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Step 5.1

Move all terms not containing $y$ to the right side of the equation.

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Step 5.1.1

Subtract $\frac{5}{3}$ from both sides of the equation.

$- \frac{y}{6} = 1 - \frac{5}{3}$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.1.2

Write $1$ as a fraction with a common denominator.

$- \frac{y}{6} = \frac{3}{3} - \frac{5}{3}$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.1.3

Combine the numerators over the common denominator.

$- \frac{y}{6} = \frac{3 - 5}{3}$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.1.4

Subtract $5$ from $3$ .

$- \frac{y}{6} = \frac{- 2}{3}$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.1.5

Move the negative in front of the fraction.

$- \frac{y}{6} = - \frac{2}{3}$

$x = \frac{1}{3} - \frac{5 y}{6}$

$- \frac{y}{6} = - \frac{2}{3}$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.2

Multiply both sides of the equation by $- 6$ .

$- 6 (- \frac{y}{6}) = - 6 (- \frac{2}{3})$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $- 6 (- \frac{y}{6})$ .

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Step 5.3.1.1.1

Cancel the common factor of $6$ .

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Step 5.3.1.1.1.1

Move the leading negative in $- \frac{y}{6}$ into the numerator.

$- 6 \frac{- y}{6} = - 6 (- \frac{2}{3})$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.3.1.1.1.2

Factor $6$ out of $- 6$ .

$6 (- 1) (\frac{- y}{6}) = - 6 (- \frac{2}{3})$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.3.1.1.1.3

Cancel the common factor.

$6 \cdot (- 1 \frac{- y}{6}) = - 6 (- \frac{2}{3})$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.3.1.1.1.4

Rewrite the expression.

$y = - 6 (- \frac{2}{3})$

$x = \frac{1}{3} - \frac{5 y}{6}$

$y = - 6 (- \frac{2}{3})$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.3.1.1.2

Multiply.

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Step 5.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 y = - 6 (- \frac{2}{3})$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.3.1.1.2.2

Multiply $y$ by $1$ .

$y = - 6 (- \frac{2}{3})$

$x = \frac{1}{3} - \frac{5 y}{6}$

$y = - 6 (- \frac{2}{3})$

$x = \frac{1}{3} - \frac{5 y}{6}$

$y = - 6 (- \frac{2}{3})$

$x = \frac{1}{3} - \frac{5 y}{6}$

$y = - 6 (- \frac{2}{3})$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Simplify $- 6 (- \frac{2}{3})$ .

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Step 5.3.2.1.1

Cancel the common factor of $3$ .

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Step 5.3.2.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$y = - 6 (\frac{- 2}{3})$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.3.2.1.1.2

Factor $3$ out of $- 6$ .

$y = 3 (- 2) (\frac{- 2}{3})$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.3.2.1.1.3

Cancel the common factor.

$y = 3 \cdot (- 2 (\frac{- 2}{3}))$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.3.2.1.1.4

Rewrite the expression.

$y = - 2 \cdot - 2$

$x = \frac{1}{3} - \frac{5 y}{6}$

$y = - 2 \cdot - 2$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 5.3.2.1.2

Multiply $- 2$ by $- 2$ .

$y = 4$

$x = \frac{1}{3} - \frac{5 y}{6}$

$y = 4$

$x = \frac{1}{3} - \frac{5 y}{6}$

$y = 4$

$x = \frac{1}{3} - \frac{5 y}{6}$

$y = 4$

$x = \frac{1}{3} - \frac{5 y}{6}$

$y = 4$

$x = \frac{1}{3} - \frac{5 y}{6}$

Step 6

Replace all occurrences of $y$ with $4$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = \frac{1}{3} - \frac{5 y}{6}$ with $4$ .

$x = \frac{1}{3} - \frac{5 (4)}{6}$

$y = 4$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{1}{3} - \frac{5 (4)}{6}$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Cancel the common factor of $4$ and $6$ .

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Step 6.2.1.1.1.1

Factor $2$ out of $5 (4)$ .

$x = \frac{1}{3} - \frac{2 (5 \cdot (2))}{6}$

$y = 4$

Step 6.2.1.1.1.2

Cancel the common factors.

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Step 6.2.1.1.1.2.1

Factor $2$ out of $6$ .

$x = \frac{1}{3} - \frac{2 (5 \cdot (2))}{2 (3)}$

$y = 4$

Step 6.2.1.1.1.2.2

Cancel the common factor.

$x = \frac{1}{3} - \frac{2 (5 \cdot (2))}{2 \cdot 3}$

$y = 4$

Step 6.2.1.1.1.2.3

Rewrite the expression.

$x = \frac{1}{3} - \frac{5 \cdot (2)}{3}$

$y = 4$

$x = \frac{1}{3} - \frac{5 \cdot (2)}{3}$

$y = 4$

$x = \frac{1}{3} - \frac{5 \cdot (2)}{3}$

$y = 4$

Step 6.2.1.1.2

Multiply $5$ by $2$ .

$x = \frac{1}{3} - \frac{10}{3}$

$y = 4$

$x = \frac{1}{3} - \frac{10}{3}$

$y = 4$

Step 6.2.1.2

Combine fractions.

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Step 6.2.1.2.1

Combine the numerators over the common denominator.

$x = \frac{1 - 10}{3}$

$y = 4$

Step 6.2.1.2.2

Simplify the expression.

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Step 6.2.1.2.2.1

Subtract $10$ from $1$ .

$x = \frac{- 9}{3}$

$y = 4$

Step 6.2.1.2.2.2

Divide $- 9$ by $3$ .

$x = - 3$

$y = 4$

$x = - 3$

$y = 4$

$x = - 3$

$y = 4$

$x = - 3$

$y = 4$

$x = - 3$

$y = 4$

$x = - 3$

$y = 4$

Step 7

List all of the solutions.

$x = - 3, y = 4$