Linear Algebra Examples

$9 x + 2 y = 30$ , $3 x - 4 y = 24$

Step 1

Solve for $x$ in $9 x + 2 y = 30$ .

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Step 1.1

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

$9 x = 30 - 2 y$

$3 x - 4 y = 24$

Step 1.2

Divide each term in $9 x = 30 - 2 y$ by $9$ and simplify.

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Step 1.2.1

Divide each term in $9 x = 30 - 2 y$ by $9$ .

$\frac{9 x}{9} = \frac{30}{9} + \frac{- 2 y}{9}$

$3 x - 4 y = 24$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $9$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{30}{9} + \frac{- 2 y}{9}$

$3 x - 4 y = 24$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{30}{9} + \frac{- 2 y}{9}$

$3 x - 4 y = 24$

$x = \frac{30}{9} + \frac{- 2 y}{9}$

$3 x - 4 y = 24$

$x = \frac{30}{9} + \frac{- 2 y}{9}$

$3 x - 4 y = 24$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Cancel the common factor of $30$ and $9$ .

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Step 1.2.3.1.1.1

Factor $3$ out of $30$ .

$x = \frac{3 (10)}{9} + \frac{- 2 y}{9}$

$3 x - 4 y = 24$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $3$ out of $9$ .

$x = \frac{3 \cdot 10}{3 \cdot 3} + \frac{- 2 y}{9}$

$3 x - 4 y = 24$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{3 \cdot 10}{3 \cdot 3} + \frac{- 2 y}{9}$

$3 x - 4 y = 24$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{10}{3} + \frac{- 2 y}{9}$

$3 x - 4 y = 24$

$x = \frac{10}{3} + \frac{- 2 y}{9}$

$3 x - 4 y = 24$

$x = \frac{10}{3} + \frac{- 2 y}{9}$

$3 x - 4 y = 24$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = \frac{10}{3} - \frac{2 y}{9}$

$3 x - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$3 x - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$3 x - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$3 x - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$3 x - 4 y = 24$

Step 2

Replace all occurrences of $x$ with $\frac{10}{3} - \frac{2 y}{9}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x - 4 y = 24$ with $\frac{10}{3} - \frac{2 y}{9}$ .

$3 (\frac{10}{3} - \frac{2 y}{9}) - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (\frac{10}{3} - \frac{2 y}{9}) - 4 y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$3 (\frac{10}{3}) + 3 (- \frac{2 y}{9}) - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$3 (\frac{10}{3}) + 3 (- \frac{2 y}{9}) - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.1.2.2

Rewrite the expression.

$10 + 3 (- \frac{2 y}{9}) - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$10 + 3 (- \frac{2 y}{9}) - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.1.3

Cancel the common factor of $3$ .

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Step 2.2.1.1.3.1

Move the leading negative in $- \frac{2 y}{9}$ into the numerator.

$10 + 3 (\frac{- 2 y}{9}) - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.1.3.2

Factor $3$ out of $9$ .

$10 + 3 (\frac{- 2 y}{3 (3)}) - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.1.3.3

Cancel the common factor.

$10 + 3 (\frac{- 2 y}{3 \cdot 3}) - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.1.3.4

Rewrite the expression.

$10 + \frac{- 2 y}{3} - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$10 + \frac{- 2 y}{3} - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.1.4

Move the negative in front of the fraction.

$10 - \frac{2 y}{3} - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$10 - \frac{2 y}{3} - 4 y = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.2

To write $- 4 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$10 - \frac{2 y}{3} - 4 y \cdot \frac{3}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $- 4 y$ and $\frac{3}{3}$ .

$10 - \frac{2 y}{3} + \frac{- 4 y \cdot 3}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$10 + \frac{- 2 y - 4 y \cdot 3}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$10 + \frac{- 2 y - 4 y \cdot 3}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.4

Simplify each term.

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Step 2.2.1.4.1

Simplify the numerator.

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Step 2.2.1.4.1.1

Factor $2 y$ out of $- 2 y - 4 y \cdot 3$ .

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Step 2.2.1.4.1.1.1

Factor $2 y$ out of $- 2 y$ .

$10 + \frac{2 y (- 1) - 4 y \cdot 3}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.4.1.1.2

Factor $2 y$ out of $- 4 y \cdot 3$ .

$10 + \frac{2 y (- 1) + 2 y (- 2 \cdot 3)}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.4.1.1.3

Factor $2 y$ out of $2 y (- 1) + 2 y (- 2 \cdot 3)$ .

$10 + \frac{2 y (- 1 - 2 \cdot 3)}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$10 + \frac{2 y (- 1 - 2 \cdot 3)}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.4.1.2

Multiply $- 2$ by $3$ .

$10 + \frac{2 y (- 1 - 6)}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.4.1.3

Subtract $6$ from $- 1$ .

$10 + \frac{2 y \cdot - 7}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.4.1.4

Multiply $- 7$ by $2$ .

$10 + \frac{- 14 y}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$10 + \frac{- 14 y}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 2.2.1.4.2

Move the negative in front of the fraction.

$10 - \frac{14 y}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$10 - \frac{14 y}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$10 - \frac{14 y}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$10 - \frac{14 y}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

$10 - \frac{14 y}{3} = 24$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3

Solve for $y$ in $10 - \frac{14 y}{3} = 24$ .

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Step 3.1

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

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Step 3.1.1

Subtract $10$ from both sides of the equation.

$- \frac{14 y}{3} = 24 - 10$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.1.2

Subtract $10$ from $24$ .

$- \frac{14 y}{3} = 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

$- \frac{14 y}{3} = 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.2

Multiply both sides of the equation by $- \frac{3}{14}$ .

$- \frac{3}{14} \cdot (- \frac{14 y}{3}) = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $- \frac{3}{14} (- \frac{14 y}{3})$ .

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Step 3.3.1.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1.1

Move the leading negative in $- \frac{3}{14}$ into the numerator.

$\frac{- 3}{14} \cdot (- \frac{14 y}{3}) = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.3.1.1.1.2

Move the leading negative in $- \frac{14 y}{3}$ into the numerator.

$\frac{- 3}{14} \cdot \frac{- 14 y}{3} = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.3.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{14} \cdot \frac{- 14 y}{3} = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.3.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{14} \cdot \frac{- 14 y}{3} = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{14} \cdot (- 14 y) = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

$\frac{- 1}{14} \cdot (- 14 y) = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.3.1.1.2

Cancel the common factor of $14$ .

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Step 3.3.1.1.2.1

Factor $14$ out of $- 14 y$ .

$\frac{- 1}{14} \cdot (14 (- y)) = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{14} \cdot (14 (- y)) = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

$y = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.3.1.1.3

Multiply.

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Step 3.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.3.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

$y = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

$y = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

$y = - \frac{3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Cancel the common factor of $14$ .

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Step 3.3.2.1.1

Move the leading negative in $- \frac{3}{14}$ into the numerator.

$y = \frac{- 3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.3.2.1.2

Cancel the common factor.

$y = \frac{- 3}{14} \cdot 14$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 3.3.2.1.3

Rewrite the expression.

$y = - 3$

$x = \frac{10}{3} - \frac{2 y}{9}$

$y = - 3$

$x = \frac{10}{3} - \frac{2 y}{9}$

$y = - 3$

$x = \frac{10}{3} - \frac{2 y}{9}$

$y = - 3$

$x = \frac{10}{3} - \frac{2 y}{9}$

$y = - 3$

$x = \frac{10}{3} - \frac{2 y}{9}$

Step 4

Replace all occurrences of $y$ with $- 3$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{10}{3} - \frac{2 y}{9}$ with $- 3$ .

$x = \frac{10}{3} - \frac{2 (- 3)}{9}$

$y = - 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{10}{3} - \frac{2 (- 3)}{9}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $- 3$ and $9$ .

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Step 4.2.1.1.1.1

Factor $3$ out of $2 (- 3)$ .

$x = \frac{10}{3} - \frac{3 (2 \cdot (- 1))}{9}$

$y = - 3$

Step 4.2.1.1.1.2

Cancel the common factors.

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Step 4.2.1.1.1.2.1

Factor $3$ out of $9$ .

$x = \frac{10}{3} - \frac{3 (2 \cdot (- 1))}{3 (3)}$

$y = - 3$

Step 4.2.1.1.1.2.2

Cancel the common factor.

$x = \frac{10}{3} - \frac{3 (2 \cdot (- 1))}{3 \cdot 3}$

$y = - 3$

Step 4.2.1.1.1.2.3

Rewrite the expression.

$x = \frac{10}{3} - \frac{2 \cdot (- 1)}{3}$

$y = - 3$

$x = \frac{10}{3} - \frac{2 \cdot (- 1)}{3}$

$y = - 3$

$x = \frac{10}{3} - \frac{2 \cdot (- 1)}{3}$

$y = - 3$

Step 4.2.1.1.2

Multiply $2$ by $- 1$ .

$x = \frac{10}{3} - \frac{- 2}{3}$

$y = - 3$

Step 4.2.1.1.3

Move the negative in front of the fraction.

$x = \frac{10}{3} + \frac{2}{3}$

$y = - 3$

Step 4.2.1.1.4

Multiply $- - \frac{2}{3}$ .

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Step 4.2.1.1.4.1

Multiply $- 1$ by $- 1$ .

$x = \frac{10}{3} + 1 (\frac{2}{3})$

$y = - 3$

Step 4.2.1.1.4.2

Multiply $\frac{2}{3}$ by $1$ .

$x = \frac{10}{3} + \frac{2}{3}$

$y = - 3$

$x = \frac{10}{3} + \frac{2}{3}$

$y = - 3$

$x = \frac{10}{3} + \frac{2}{3}$

$y = - 3$

Step 4.2.1.2

Combine fractions.

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Step 4.2.1.2.1

Combine the numerators over the common denominator.

$x = \frac{10 + 2}{3}$

$y = - 3$

Step 4.2.1.2.2

Simplify the expression.

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Step 4.2.1.2.2.1

Add $10$ and $2$ .

$x = \frac{12}{3}$

$y = - 3$

Step 4.2.1.2.2.2

Divide $12$ by $3$ .

$x = 4$

$y = - 3$

$x = 4$

$y = - 3$

$x = 4$

$y = - 3$

$x = 4$

$y = - 3$

$x = 4$

$y = - 3$

$x = 4$

$y = - 3$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(4, - 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(4, - 3)$

Equation Form:

$x = 4, y = - 3$

Step 7