Linear Algebra Examples

$[\begin{array}{c} - 3 & 1 & 2 \\ 4 & 2 & - 4 \\ 0 & 1 & 3 \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 5 \\ - 4 \\ 3 \end{array}]$

Step 1

Multiply $[\begin{array}{c} - 3 & 1 & 2 \\ 4 & 2 & - 4 \\ 0 & 1 & 3 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}]$ .

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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 $3 \times 3$ and the second matrix is $3 \times 1$ .

Step 1.2

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

$[\begin{array}{c} - 3 x + 1 y + 2 z \\ 4 x + 2 y - 4 z \\ 0 x + 1 y + 3 z \end{array}] = [\begin{array}{c} 5 \\ - 4 \\ 3 \end{array}]$

Step 1.3

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

$[\begin{array}{c} - 3 x + y + 2 z \\ 4 x + 2 y - 4 z \\ y + 3 z \end{array}] = [\begin{array}{c} 5 \\ - 4 \\ 3 \end{array}]$

Step 2

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

$- 3 x + y + 2 z = 5$

$4 x + 2 y - 4 z = - 4$

$y + 3 z = 3$

Step 3

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

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Step 3.1

Add $3 x$ to both sides of the equation.

$y + 2 z = 5 + 3 x$

$4 x + 2 y - 4 z = - 4$

$y + 3 z = 3$

Step 3.2

Subtract $2 z$ from both sides of the equation.

$y = 5 + 3 x - 2 z$

$4 x + 2 y - 4 z = - 4$

$y + 3 z = 3$

$y = 5 + 3 x - 2 z$

$4 x + 2 y - 4 z = - 4$

$y + 3 z = 3$

Step 4

Replace all occurrences of $y$ with $5 + 3 x - 2 z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $4 x + 2 y - 4 z = - 4$ with $5 + 3 x - 2 z$ .

$4 x + 2 (5 + 3 x - 2 z) - 4 z = - 4$

$y = 5 + 3 x - 2 z$

$y + 3 z = 3$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $4 x + 2 (5 + 3 x - 2 z) - 4 z$ .

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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.

$4 x + 2 \cdot 5 + 2 (3 x) + 2 (- 2 z) - 4 z = - 4$

$y = 5 + 3 x - 2 z$

$y + 3 z = 3$

Step 4.2.1.1.2

Simplify.

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Step 4.2.1.1.2.1

Multiply $2$ by $5$ .

$4 x + 10 + 2 (3 x) + 2 (- 2 z) - 4 z = - 4$

$y = 5 + 3 x - 2 z$

$y + 3 z = 3$

Step 4.2.1.1.2.2

Multiply $3$ by $2$ .

$4 x + 10 + 6 x + 2 (- 2 z) - 4 z = - 4$

$y = 5 + 3 x - 2 z$

$y + 3 z = 3$

Step 4.2.1.1.2.3

Multiply $- 2$ by $2$ .

$4 x + 10 + 6 x - 4 z - 4 z = - 4$

$y = 5 + 3 x - 2 z$

$y + 3 z = 3$

$4 x + 10 + 6 x - 4 z - 4 z = - 4$

$y = 5 + 3 x - 2 z$

$y + 3 z = 3$

$4 x + 10 + 6 x - 4 z - 4 z = - 4$

$y = 5 + 3 x - 2 z$

$y + 3 z = 3$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Add $4 x$ and $6 x$ .

$10 x + 10 - 4 z - 4 z = - 4$

$y = 5 + 3 x - 2 z$

$y + 3 z = 3$

Step 4.2.1.2.2

Subtract $4 z$ from $- 4 z$ .

$10 x + 10 - 8 z = - 4$

$y = 5 + 3 x - 2 z$

$y + 3 z = 3$

$10 x + 10 - 8 z = - 4$

$y = 5 + 3 x - 2 z$

$y + 3 z = 3$

$10 x + 10 - 8 z = - 4$

$y = 5 + 3 x - 2 z$

$y + 3 z = 3$

$10 x + 10 - 8 z = - 4$

$y = 5 + 3 x - 2 z$

$y + 3 z = 3$

Step 4.3

Replace all occurrences of $y$ in $y + 3 z = 3$ with $5 + 3 x - 2 z$ .

$(5 + 3 x - 2 z) + 3 z = 3$

$10 x + 10 - 8 z = - 4$

$y = 5 + 3 x - 2 z$

Step 4.4

Simplify the left side.

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Step 4.4.1

Add $- 2 z$ and $3 z$ .

$5 + 3 x + z = 3$

$10 x + 10 - 8 z = - 4$

$y = 5 + 3 x - 2 z$

$5 + 3 x + z = 3$

$10 x + 10 - 8 z = - 4$

$y = 5 + 3 x - 2 z$

$5 + 3 x + z = 3$

$10 x + 10 - 8 z = - 4$

$y = 5 + 3 x - 2 z$

Step 5

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

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Step 5.1

Subtract $5$ from both sides of the equation.

$3 x + z = 3 - 5$

$10 x + 10 - 8 z = - 4$

$y = 5 + 3 x - 2 z$

Step 5.2

Subtract $3 x$ from both sides of the equation.

$z = 3 - 5 - 3 x$

$10 x + 10 - 8 z = - 4$

$y = 5 + 3 x - 2 z$

Step 5.3

Subtract $5$ from $3$ .

$z = - 2 - 3 x$

$10 x + 10 - 8 z = - 4$

$y = 5 + 3 x - 2 z$

$z = - 2 - 3 x$

$10 x + 10 - 8 z = - 4$

$y = 5 + 3 x - 2 z$

Step 6

Replace all occurrences of $z$ with $- 2 - 3 x$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $10 x + 10 - 8 z = - 4$ with $- 2 - 3 x$ .

$10 x + 10 - 8 (- 2 - 3 x) = - 4$

$z = - 2 - 3 x$

$y = 5 + 3 x - 2 z$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify $10 x + 10 - 8 (- 2 - 3 x)$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Apply the distributive property.

$10 x + 10 - 8 \cdot - 2 - 8 (- 3 x) = - 4$

$z = - 2 - 3 x$

$y = 5 + 3 x - 2 z$

Step 6.2.1.1.2

Multiply $- 8$ by $- 2$ .

$10 x + 10 + 16 - 8 (- 3 x) = - 4$

$z = - 2 - 3 x$

$y = 5 + 3 x - 2 z$

Step 6.2.1.1.3

Multiply $- 3$ by $- 8$ .

$10 x + 10 + 16 + 24 x = - 4$

$z = - 2 - 3 x$

$y = 5 + 3 x - 2 z$

$10 x + 10 + 16 + 24 x = - 4$

$z = - 2 - 3 x$

$y = 5 + 3 x - 2 z$

Step 6.2.1.2

Simplify by adding terms.

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Step 6.2.1.2.1

Add $10 x$ and $24 x$ .

$34 x + 10 + 16 = - 4$

$z = - 2 - 3 x$

$y = 5 + 3 x - 2 z$

Step 6.2.1.2.2

Add $10$ and $16$ .

$34 x + 26 = - 4$

$z = - 2 - 3 x$

$y = 5 + 3 x - 2 z$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

$y = 5 + 3 x - 2 z$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

$y = 5 + 3 x - 2 z$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

$y = 5 + 3 x - 2 z$

Step 6.3

Replace all occurrences of $z$ in $y = 5 + 3 x - 2 z$ with $- 2 - 3 x$ .

$y = 5 + 3 x - 2 (- 2 - 3 x)$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $5 + 3 x - 2 (- 2 - 3 x)$ .

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Step 6.4.1.1

Simplify each term.

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Step 6.4.1.1.1

Apply the distributive property.

$y = 5 + 3 x - 2 \cdot - 2 - 2 (- 3 x)$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

Step 6.4.1.1.2

Multiply $- 2$ by $- 2$ .

$y = 5 + 3 x + 4 - 2 (- 3 x)$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

Step 6.4.1.1.3

Multiply $- 3$ by $- 2$ .

$y = 5 + 3 x + 4 + 6 x$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

$y = 5 + 3 x + 4 + 6 x$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

Step 6.4.1.2

Simplify by adding terms.

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Step 6.4.1.2.1

Add $5$ and $4$ .

$y = 3 x + 9 + 6 x$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

Step 6.4.1.2.2

Add $3 x$ and $6 x$ .

$y = 9 x + 9$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

$y = 9 x + 9$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

$y = 9 x + 9$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

$y = 9 x + 9$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

$y = 9 x + 9$

$34 x + 26 = - 4$

$z = - 2 - 3 x$

Step 7

Solve for $x$ in $34 x + 26 = - 4$ .

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Step 7.1

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

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Step 7.1.1

Subtract $26$ from both sides of the equation.

$34 x = - 4 - 26$

$y = 9 x + 9$

$z = - 2 - 3 x$

Step 7.1.2

Subtract $26$ from $- 4$ .

$34 x = - 30$

$y = 9 x + 9$

$z = - 2 - 3 x$

$34 x = - 30$

$y = 9 x + 9$

$z = - 2 - 3 x$

Step 7.2

Divide each term in $34 x = - 30$ by $34$ and simplify.

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Step 7.2.1

Divide each term in $34 x = - 30$ by $34$ .

$\frac{34 x}{34} = \frac{- 30}{34}$

$y = 9 x + 9$

$z = - 2 - 3 x$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $34$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{34 x}{34} = \frac{- 30}{34}$

$y = 9 x + 9$

$z = - 2 - 3 x$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 30}{34}$

$y = 9 x + 9$

$z = - 2 - 3 x$

$x = \frac{- 30}{34}$

$y = 9 x + 9$

$z = - 2 - 3 x$

$x = \frac{- 30}{34}$

$y = 9 x + 9$

$z = - 2 - 3 x$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Cancel the common factor of $- 30$ and $34$ .

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Step 7.2.3.1.1

Factor $2$ out of $- 30$ .

$x = \frac{2 (- 15)}{34}$

$y = 9 x + 9$

$z = - 2 - 3 x$

Step 7.2.3.1.2

Cancel the common factors.

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Step 7.2.3.1.2.1

Factor $2$ out of $34$ .

$x = \frac{2 \cdot - 15}{2 \cdot 17}$

$y = 9 x + 9$

$z = - 2 - 3 x$

Step 7.2.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 15}{2 \cdot 17}$

$y = 9 x + 9$

$z = - 2 - 3 x$

Step 7.2.3.1.2.3

Rewrite the expression.

$x = \frac{- 15}{17}$

$y = 9 x + 9$

$z = - 2 - 3 x$

$x = \frac{- 15}{17}$

$y = 9 x + 9$

$z = - 2 - 3 x$

$x = \frac{- 15}{17}$

$y = 9 x + 9$

$z = - 2 - 3 x$

Step 7.2.3.2

Move the negative in front of the fraction.

$x = - \frac{15}{17}$

$y = 9 x + 9$

$z = - 2 - 3 x$

$x = - \frac{15}{17}$

$y = 9 x + 9$

$z = - 2 - 3 x$

$x = - \frac{15}{17}$

$y = 9 x + 9$

$z = - 2 - 3 x$

$x = - \frac{15}{17}$

$y = 9 x + 9$

$z = - 2 - 3 x$

Step 8

Replace all occurrences of $x$ with $- \frac{15}{17}$ in each equation.

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Step 8.1

Replace all occurrences of $x$ in $y = 9 x + 9$ with $- \frac{15}{17}$ .

$y = 9 (- \frac{15}{17}) + 9$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $9 (- \frac{15}{17}) + 9$ .

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Step 8.2.1.1

Simplify each term.

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Step 8.2.1.1.1

Multiply $9 (- \frac{15}{17})$ .

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Step 8.2.1.1.1.1

Multiply $- 1$ by $9$ .

$y = - 9 (\frac{15}{17}) + 9$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

Step 8.2.1.1.1.2

Combine $- 9$ and $\frac{15}{17}$ .

$y = \frac{- 9 \cdot 15}{17} + 9$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

Step 8.2.1.1.1.3

Multiply $- 9$ by $15$ .

$y = \frac{- 135}{17} + 9$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

$y = \frac{- 135}{17} + 9$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

Step 8.2.1.1.2

Move the negative in front of the fraction.

$y = - \frac{135}{17} + 9$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

$y = - \frac{135}{17} + 9$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

Step 8.2.1.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$y = - \frac{135}{17} + 9 \cdot \frac{17}{17}$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

Step 8.2.1.3

Combine $9$ and $\frac{17}{17}$ .

$y = - \frac{135}{17} + \frac{9 \cdot 17}{17}$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

Step 8.2.1.4

Combine the numerators over the common denominator.

$y = \frac{- 135 + 9 \cdot 17}{17}$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

Step 8.2.1.5

Simplify the numerator.

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Step 8.2.1.5.1

Multiply $9$ by $17$ .

$y = \frac{- 135 + 153}{17}$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

Step 8.2.1.5.2

Add $- 135$ and $153$ .

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

$z = - 2 - 3 x$

Step 8.3

Replace all occurrences of $x$ in $z = - 2 - 3 x$ with $- \frac{15}{17}$ .

$z = - 2 - 3 (- \frac{15}{17})$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

Step 8.4

Simplify the right side.

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Step 8.4.1

Simplify $- 2 - 3 (- \frac{15}{17})$ .

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Step 8.4.1.1

Multiply $- 3 (- \frac{15}{17})$ .

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Step 8.4.1.1.1

Multiply $- 1$ by $- 3$ .

$z = - 2 + 3 (\frac{15}{17})$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

Step 8.4.1.1.2

Combine $3$ and $\frac{15}{17}$ .

$z = - 2 + \frac{3 \cdot 15}{17}$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

Step 8.4.1.1.3

Multiply $3$ by $15$ .

$z = - 2 + \frac{45}{17}$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

$z = - 2 + \frac{45}{17}$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

Step 8.4.1.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$z = - 2 \cdot \frac{17}{17} + \frac{45}{17}$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

Step 8.4.1.3

Combine $- 2$ and $\frac{17}{17}$ .

$z = \frac{- 2 \cdot 17}{17} + \frac{45}{17}$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

Step 8.4.1.4

Combine the numerators over the common denominator.

$z = \frac{- 2 \cdot 17 + 45}{17}$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

Step 8.4.1.5

Simplify the numerator.

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Step 8.4.1.5.1

Multiply $- 2$ by $17$ .

$z = \frac{- 34 + 45}{17}$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

Step 8.4.1.5.2

Add $- 34$ and $45$ .

$z = \frac{11}{17}$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

$z = \frac{11}{17}$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

$z = \frac{11}{17}$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

$z = \frac{11}{17}$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

$z = \frac{11}{17}$

$y = \frac{18}{17}$

$x = - \frac{15}{17}$

Step 9

List all of the solutions.

$z = \frac{11}{17}, y = \frac{18}{17}, x = - \frac{15}{17}$