Linear Algebra Examples

$- 8 x - 10 y = 24$ , $6 x + 5 y = 2$

Step 1

Solve for $x$ in $- 8 x - 10 y = 24$ .

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Step 1.1

Add $10 y$ to both sides of the equation.

$- 8 x = 24 + 10 y$

$6 x + 5 y = 2$

Step 1.2

Divide each term in $- 8 x = 24 + 10 y$ by $- 8$ and simplify.

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Step 1.2.1

Divide each term in $- 8 x = 24 + 10 y$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{24}{- 8} + \frac{10 y}{- 8}$

$6 x + 5 y = 2$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 8$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{24}{- 8} + \frac{10 y}{- 8}$

$6 x + 5 y = 2$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{24}{- 8} + \frac{10 y}{- 8}$

$6 x + 5 y = 2$

$x = \frac{24}{- 8} + \frac{10 y}{- 8}$

$6 x + 5 y = 2$

$x = \frac{24}{- 8} + \frac{10 y}{- 8}$

$6 x + 5 y = 2$

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

Divide $24$ by $- 8$ .

$x = - 3 + \frac{10 y}{- 8}$

$6 x + 5 y = 2$

Step 1.2.3.1.2

Cancel the common factor of $10$ and $- 8$ .

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Step 1.2.3.1.2.1

Factor $2$ out of $10 y$ .

$x = - 3 + \frac{2 (5 y)}{- 8}$

$6 x + 5 y = 2$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $- 8$ .

$x = - 3 + \frac{2 (5 y)}{2 (- 4)}$

$6 x + 5 y = 2$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = - 3 + \frac{2 (5 y)}{2 \cdot - 4}$

$6 x + 5 y = 2$

Step 1.2.3.1.2.2.3

Rewrite the expression.

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

$6 x + 5 y = 2$

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

$6 x + 5 y = 2$

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

$6 x + 5 y = 2$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$x = - 3 - \frac{5 y}{4}$

$6 x + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

$6 x + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

$6 x + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

$6 x + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

$6 x + 5 y = 2$

Step 2

Replace all occurrences of $x$ with $- 3 - \frac{5 y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $6 x + 5 y = 2$ with $- 3 - \frac{5 y}{4}$ .

$6 (- 3 - \frac{5 y}{4}) + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$6 \cdot - 3 + 6 (- \frac{5 y}{4}) + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.1.2

Multiply $6$ by $- 3$ .

$- 18 + 6 (- \frac{5 y}{4}) + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.1.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.3.1

Move the leading negative in $- \frac{5 y}{4}$ into the numerator.

$- 18 + 6 (\frac{- 5 y}{4}) + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.1.3.2

Factor $2$ out of $6$ .

$- 18 + 2 (3) (\frac{- 5 y}{4}) + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.1.3.3

Factor $2$ out of $4$ .

$- 18 + 2 \cdot (3 (\frac{- 5 y}{2 \cdot 2})) + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.1.3.4

Cancel the common factor.

$- 18 + 2 \cdot (3 (\frac{- 5 y}{2 \cdot 2})) + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.1.3.5

Rewrite the expression.

$- 18 + 3 (\frac{- 5 y}{2}) + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

$- 18 + 3 (\frac{- 5 y}{2}) + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.1.4

Combine $3$ and $\frac{- 5 y}{2}$ .

$- 18 + \frac{3 (- 5 y)}{2} + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.1.5

Multiply $- 5$ by $3$ .

$- 18 + \frac{- 15 y}{2} + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.1.6

Move the negative in front of the fraction.

$- 18 - \frac{15 y}{2} + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

$- 18 - \frac{15 y}{2} + 5 y = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.2

To write $5 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 18 - \frac{15 y}{2} + 5 y \cdot \frac{2}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $5 y$ and $\frac{2}{2}$ .

$- 18 - \frac{15 y}{2} + \frac{5 y \cdot 2}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$- 18 + \frac{- 15 y + 5 y \cdot 2}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

$- 18 + \frac{- 15 y + 5 y \cdot 2}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

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 $5 y$ out of $- 15 y + 5 y \cdot 2$ .

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Step 2.2.1.4.1.1.1

Factor $5 y$ out of $- 15 y$ .

$- 18 + \frac{5 y (- 3) + 5 y \cdot 2}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.4.1.1.2

Factor $5 y$ out of $5 y \cdot 2$ .

$- 18 + \frac{5 y (- 3) + 5 y (2)}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.4.1.1.3

Factor $5 y$ out of $5 y (- 3) + 5 y (2)$ .

$- 18 + \frac{5 y (- 3 + 2)}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

$- 18 + \frac{5 y (- 3 + 2)}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.4.1.2

Add $- 3$ and $2$ .

$- 18 + \frac{5 y \cdot - 1}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.4.1.3

Multiply $- 1$ by $5$ .

$- 18 + \frac{- 5 y}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

$- 18 + \frac{- 5 y}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

Step 2.2.1.4.2

Move the negative in front of the fraction.

$- 18 - \frac{5 y}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

$- 18 - \frac{5 y}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

$- 18 - \frac{5 y}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

$- 18 - \frac{5 y}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

$- 18 - \frac{5 y}{2} = 2$

$x = - 3 - \frac{5 y}{4}$

Step 3

Solve for $y$ in $- 18 - \frac{5 y}{2} = 2$ .

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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

Add $18$ to both sides of the equation.

$- \frac{5 y}{2} = 2 + 18$

$x = - 3 - \frac{5 y}{4}$

Step 3.1.2

Add $2$ and $18$ .

$- \frac{5 y}{2} = 20$

$x = - 3 - \frac{5 y}{4}$

$- \frac{5 y}{2} = 20$

$x = - 3 - \frac{5 y}{4}$

Step 3.2

Multiply both sides of the equation by $- \frac{2}{5}$ .

$- \frac{2}{5} \cdot (- \frac{5 y}{2}) = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

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 $- \frac{2}{5} (- \frac{5 y}{2})$ .

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Step 3.3.1.1.1

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.1

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$\frac{- 2}{5} \cdot (- \frac{5 y}{2}) = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.1.1.1.2

Move the leading negative in $- \frac{5 y}{2}$ into the numerator.

$\frac{- 2}{5} \cdot \frac{- 5 y}{2} = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{5} \cdot \frac{- 5 y}{2} = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{5} \cdot \frac{- 5 y}{2} = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{5} \cdot (- 5 y) = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

$\frac{- 1}{5} \cdot (- 5 y) = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.1.1.2

Cancel the common factor of $5$ .

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Step 3.3.1.1.2.1

Factor $5$ out of $- 5 y$ .

$\frac{- 1}{5} \cdot (5 (- y)) = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{5} \cdot (5 (- y)) = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

$y = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.1.1.3

Multiply.

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Step 3.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

$y = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

$y = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

$y = - \frac{2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \frac{2}{5} \cdot 20$ .

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Step 3.3.2.1.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1.1

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$y = \frac{- 2}{5} \cdot 20$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.2.1.1.2

Factor $5$ out of $20$ .

$y = \frac{- 2}{5} \cdot (5 (4))$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.2.1.1.3

Cancel the common factor.

$y = \frac{- 2}{5} \cdot (5 \cdot 4)$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.2.1.1.4

Rewrite the expression.

$y = - 2 \cdot 4$

$x = - 3 - \frac{5 y}{4}$

$y = - 2 \cdot 4$

$x = - 3 - \frac{5 y}{4}$

Step 3.3.2.1.2

Multiply $- 2$ by $4$ .

$y = - 8$

$x = - 3 - \frac{5 y}{4}$

$y = - 8$

$x = - 3 - \frac{5 y}{4}$

$y = - 8$

$x = - 3 - \frac{5 y}{4}$

$y = - 8$

$x = - 3 - \frac{5 y}{4}$

$y = - 8$

$x = - 3 - \frac{5 y}{4}$

Step 4

Replace all occurrences of $y$ with $- 8$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - 3 - \frac{5 y}{4}$ with $- 8$ .

$x = - 3 - \frac{5 (- 8)}{4}$

$y = - 8$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 3 - \frac{5 (- 8)}{4}$ .

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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 $- 8$ and $4$ .

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Step 4.2.1.1.1.1

Factor $4$ out of $5 (- 8)$ .

$x = - 3 - \frac{4 (5 \cdot (- 2))}{4}$

$y = - 8$

Step 4.2.1.1.1.2

Cancel the common factors.

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Step 4.2.1.1.1.2.1

Factor $4$ out of $4$ .

$x = - 3 - \frac{4 (5 \cdot (- 2))}{4 (1)}$

$y = - 8$

Step 4.2.1.1.1.2.2

Cancel the common factor.

$x = - 3 - \frac{4 (5 \cdot (- 2))}{4 \cdot 1}$

$y = - 8$

Step 4.2.1.1.1.2.3

Rewrite the expression.

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

$y = - 8$

Step 4.2.1.1.1.2.4

Divide $5 \cdot (- 2)$ by $1$ .

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

$y = - 8$

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

$y = - 8$

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

$y = - 8$

Step 4.2.1.1.2

Multiply $- (5 \cdot (- 2))$ .

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Step 4.2.1.1.2.1

Multiply $5$ by $- 2$ .

$x = - 3 + 10$

$y = - 8$

Step 4.2.1.1.2.2

Multiply $- 1$ by $- 10$ .

$x = - 3 + 10$

$y = - 8$

$x = - 3 + 10$

$y = - 8$

$x = - 3 + 10$

$y = - 8$

Step 4.2.1.2

Add $- 3$ and $10$ .

$x = 7$

$y = - 8$

$x = 7$

$y = - 8$

$x = 7$

$y = - 8$

$x = 7$

$y = - 8$

Step 5

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

$(7, - 8)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(7, - 8)$

Equation Form:

$x = 7, y = - 8$

Step 7