Linear Algebra Examples

$[\begin{array}{c} 3 x & 5 \\ - 1 & 4 x \end{array}] [\begin{array}{c} 2 y & - 3 \\ - 6 & - y \end{array}] = [\begin{array}{c} 7 & 2 \\ - 7 & 2 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 3 x & 5 \\ - 1 & 4 x \end{array}] [\begin{array}{c} 2 y & - 3 \\ - 6 & - y \end{array}]$ .

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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 $2 \times 2$ .

Step 1.2

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

$[\begin{array}{c} 3 x (2 y) + 5 \cdot - 6 & 3 x \cdot - 3 + 5 (- y) \\ - (2 y) + 4 x \cdot - 6 & - - 3 + 4 x (- y) \end{array}] = [\begin{array}{c} 7 & 2 \\ - 7 & 2 \end{array}]$

Step 1.3

Simplify each element of the matrix by multiplying out all the expressions.

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Step 1.3.1

Multiply $- 1$ by $- 3$ .

$[\begin{array}{c} 6 x y - 30 & - 9 x - 5 y \\ - 2 y - 24 x & 3 + 4 x (- y) \end{array}] = [\begin{array}{c} 7 & 2 \\ - 7 & 2 \end{array}]$

Step 1.3.2

Rewrite using the commutative property of multiplication.

$[\begin{array}{c} 6 x y - 30 & - 9 x - 5 y \\ - 2 y - 24 x & 3 + 4 \cdot - 1 x y \end{array}] = [\begin{array}{c} 7 & 2 \\ - 7 & 2 \end{array}]$

Step 1.3.3

Multiply $4$ by $- 1$ .

$[\begin{array}{c} 6 x y - 30 & - 9 x - 5 y \\ - 2 y - 24 x & 3 - 4 x y \end{array}] = [\begin{array}{c} 7 & 2 \\ - 7 & 2 \end{array}]$

Step 2

Write as a linear system of equations.

$6 x y - 30 = 7$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $x$ in $6 x y - 30 = 7$ .

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Step 3.1.1

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

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Step 3.1.1.1

Add $30$ to both sides of the equation.

$6 x y = 7 + 30$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.1.1.2

Add $7$ and $30$ .

$6 x y = 37$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

$6 x y = 37$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.1.2

Divide each term in $6 x y = 37$ by $6 y$ and simplify.

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Step 3.1.2.1

Divide each term in $6 x y = 37$ by $6 y$ .

$\frac{6 x y}{6 y} = \frac{37}{6 y}$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $6$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{6 x y}{6 y} = \frac{37}{6 y}$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.1.2.2.1.2

Rewrite the expression.

$\frac{x y}{y} = \frac{37}{6 y}$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

$\frac{x y}{y} = \frac{37}{6 y}$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.1.2.2.2

Cancel the common factor of $y$ .

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Step 3.1.2.2.2.1

Cancel the common factor.

$\frac{x y}{y} = \frac{37}{6 y}$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.1.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{37}{6 y}$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

$x = \frac{37}{6 y}$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

$x = \frac{37}{6 y}$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

$x = \frac{37}{6 y}$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

$x = \frac{37}{6 y}$

$- 9 x - 5 y = 2$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.2

Replace all occurrences of $x$ with $\frac{37}{6 y}$ in each equation.

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Step 3.2.1

Replace all occurrences of $x$ in $- 9 x - 5 y = 2$ with $\frac{37}{6 y}$ .

$- 9 \frac{37}{6 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1.1

Factor $3$ out of $- 9$ .

$3 (- 3) (\frac{37}{6 y}) - 5 y = 2$

$x = \frac{37}{6 y}$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.2.2.1.1.2

Factor $3$ out of $6 y$ .

$3 (- 3) (\frac{37}{3 (2 y)}) - 5 y = 2$

$x = \frac{37}{6 y}$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.2.2.1.1.3

Cancel the common factor.

$3 \cdot (- 3 \frac{37}{3 (2 y)}) - 5 y = 2$

$x = \frac{37}{6 y}$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.2.2.1.1.4

Rewrite the expression.

$- 3 \frac{37}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

$- 3 \frac{37}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.2.2.1.2

Combine $- 3$ and $\frac{37}{2 y}$ .

$\frac{- 3 \cdot 37}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.2.2.1.3

Multiply $- 3$ by $37$ .

$\frac{- 111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.2.2.1.4

Move the negative in front of the fraction.

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$- 2 y - 24 x = - 7$

$3 - 4 x y = 2$

Step 3.2.3

Replace all occurrences of $x$ in $- 2 y - 24 x = - 7$ with $\frac{37}{6 y}$ .

$- 2 y - 24 \frac{37}{6 y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 - 4 x y = 2$

Step 3.2.4

Simplify the left side.

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

Cancel the common factor of $6$ .

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Step 3.2.4.1.1.1

Factor $6$ out of $- 24$ .

$- 2 y + 6 (- 4) (\frac{37}{6 y}) = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 - 4 x y = 2$

Step 3.2.4.1.1.2

Factor $6$ out of $6 y$ .

$- 2 y + 6 (- 4) (\frac{37}{6 (y)}) = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 - 4 x y = 2$

Step 3.2.4.1.1.3

Cancel the common factor.

$- 2 y + 6 \cdot (- 4 \frac{37}{6 y}) = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 - 4 x y = 2$

Step 3.2.4.1.1.4

Rewrite the expression.

$- 2 y - 4 \frac{37}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 - 4 x y = 2$

$- 2 y - 4 \frac{37}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 - 4 x y = 2$

Step 3.2.4.1.2

Combine $- 4$ and $\frac{37}{y}$ .

$- 2 y + \frac{- 4 \cdot 37}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 - 4 x y = 2$

Step 3.2.4.1.3

Multiply $- 4$ by $37$ .

$- 2 y + \frac{- 148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 - 4 x y = 2$

Step 3.2.4.1.4

Move the negative in front of the fraction.

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 - 4 x y = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 - 4 x y = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 - 4 x y = 2$

Step 3.2.5

Replace all occurrences of $x$ in $3 - 4 x y = 2$ with $\frac{37}{6 y}$ .

$3 - 4 \frac{37}{6 y} \cdot y = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6

Simplify the left side.

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Step 3.2.6.1

Simplify $3 - 4 \frac{37}{6 y} \cdot y$ .

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Step 3.2.6.1.1

Simplify each term.

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Step 3.2.6.1.1.1

Cancel the common factor of $2$ .

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Step 3.2.6.1.1.1.1

Factor $2$ out of $- 4$ .

$3 + 2 (- 2) (\frac{37}{6 y}) \cdot y = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.1.1.2

Factor $2$ out of $6 y$ .

$3 + 2 (- 2) (\frac{37}{2 (3 y)}) \cdot y = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.1.1.3

Cancel the common factor.

$3 + 2 \cdot (- 2 \frac{37}{2 (3 y)}) \cdot y = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.1.1.4

Rewrite the expression.

$3 - 2 \frac{37}{3 y} \cdot y = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 - 2 \frac{37}{3 y} \cdot y = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.1.2

Combine $- 2$ and $\frac{37}{3 y}$ .

$3 + \frac{- 2 \cdot 37}{3 y} \cdot y = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.1.3

Multiply $- 2$ by $37$ .

$3 + \frac{- 74}{3 y} \cdot y = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.1.4

Cancel the common factor of $y$ .

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Step 3.2.6.1.1.4.1

Factor $y$ out of $3 y$ .

$3 + \frac{- 74}{y \cdot 3} \cdot y = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.1.4.2

Cancel the common factor.

$3 + \frac{- 74}{y \cdot 3} \cdot y = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.1.4.3

Rewrite the expression.

$3 + \frac{- 74}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 + \frac{- 74}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.1.5

Move the negative in front of the fraction.

$3 - \frac{74}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$3 - \frac{74}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 \cdot \frac{3}{3} - \frac{74}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.3

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

$\frac{3 \cdot 3}{3} - \frac{74}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.4

Combine the numerators over the common denominator.

$\frac{3 \cdot 3 - 74}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.5

Simplify the numerator.

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Step 3.2.6.1.5.1

Multiply $3$ by $3$ .

$\frac{9 - 74}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.5.2

Subtract $74$ from $9$ .

$\frac{- 65}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$\frac{- 65}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.2.6.1.6

Move the negative in front of the fraction.

$- \frac{65}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$- \frac{65}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$- \frac{65}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

$- \frac{65}{3} = 2$

$- 2 y - \frac{148}{y} = - 7$

$- \frac{111}{2 y} - 5 y = 2$

$x = \frac{37}{6 y}$

Step 3.3

Since $- \frac{65}{3} = 2$ is not true, there is no solution.

$No solution$