Linear Algebra Examples

$8 x - 6 y = - 20$ , $- 16 x + 7 y = 30$

Step 1

Solve for $x$ in $8 x - 6 y = - 20$ .

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Step 1.1

Add $6 y$ to both sides of the equation.

$8 x = - 20 + 6 y$

$- 16 x + 7 y = 30$

Step 1.2

Divide each term in $8 x = - 20 + 6 y$ by $8$ and simplify.

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Step 1.2.1

Divide each term in $8 x = - 20 + 6 y$ by $8$ .

$\frac{8 x}{8} = \frac{- 20}{8} + \frac{6 y}{8}$

$- 16 x + 7 y = 30$

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.

$\frac{8 x}{8} = \frac{- 20}{8} + \frac{6 y}{8}$

$- 16 x + 7 y = 30$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 20}{8} + \frac{6 y}{8}$

$- 16 x + 7 y = 30$

$x = \frac{- 20}{8} + \frac{6 y}{8}$

$- 16 x + 7 y = 30$

$x = \frac{- 20}{8} + \frac{6 y}{8}$

$- 16 x + 7 y = 30$

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 $- 20$ and $8$ .

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Step 1.2.3.1.1.1

Factor $4$ out of $- 20$ .

$x = \frac{4 (- 5)}{8} + \frac{6 y}{8}$

$- 16 x + 7 y = 30$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $4$ out of $8$ .

$x = \frac{4 \cdot - 5}{4 \cdot 2} + \frac{6 y}{8}$

$- 16 x + 7 y = 30$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{4 \cdot - 5}{4 \cdot 2} + \frac{6 y}{8}$

$- 16 x + 7 y = 30$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{- 5}{2} + \frac{6 y}{8}$

$- 16 x + 7 y = 30$

$x = \frac{- 5}{2} + \frac{6 y}{8}$

$- 16 x + 7 y = 30$

$x = \frac{- 5}{2} + \frac{6 y}{8}$

$- 16 x + 7 y = 30$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{5}{2} + \frac{6 y}{8}$

$- 16 x + 7 y = 30$

Step 1.2.3.1.3

Cancel the common factor of $6$ and $8$ .

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Step 1.2.3.1.3.1

Factor $2$ out of $6 y$ .

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

$- 16 x + 7 y = 30$

Step 1.2.3.1.3.2

Cancel the common factors.

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Step 1.2.3.1.3.2.1

Factor $2$ out of $8$ .

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

$- 16 x + 7 y = 30$

Step 1.2.3.1.3.2.2

Cancel the common factor.

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

$- 16 x + 7 y = 30$

Step 1.2.3.1.3.2.3

Rewrite the expression.

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

$- 16 x + 7 y = 30$

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

$- 16 x + 7 y = 30$

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

$- 16 x + 7 y = 30$

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

$- 16 x + 7 y = 30$

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

$- 16 x + 7 y = 30$

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

$- 16 x + 7 y = 30$

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

$- 16 x + 7 y = 30$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $- 16 x + 7 y = 30$ with $- \frac{5}{2} + \frac{3 y}{4}$ .

$- 16 (- \frac{5}{2} + \frac{3 y}{4}) + 7 y = 30$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 16 (- \frac{5}{2} + \frac{3 y}{4}) + 7 y$ .

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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.

$- 16 (- \frac{5}{2}) - 16 \frac{3 y}{4} + 7 y = 30$

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

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

$- 16 (\frac{- 5}{2}) - 16 \frac{3 y}{4} + 7 y = 30$

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

Step 2.2.1.1.2.2

Factor $2$ out of $- 16$ .

$2 (- 8) (\frac{- 5}{2}) - 16 \frac{3 y}{4} + 7 y = 30$

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

Step 2.2.1.1.2.3

Cancel the common factor.

$2 \cdot (- 8 (\frac{- 5}{2})) - 16 \frac{3 y}{4} + 7 y = 30$

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

Step 2.2.1.1.2.4

Rewrite the expression.

$- 8 \cdot - 5 - 16 \frac{3 y}{4} + 7 y = 30$

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

$- 8 \cdot - 5 - 16 \frac{3 y}{4} + 7 y = 30$

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

Step 2.2.1.1.3

Multiply $- 8$ by $- 5$ .

$40 - 16 \frac{3 y}{4} + 7 y = 30$

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

Step 2.2.1.1.4

Cancel the common factor of $4$ .

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Step 2.2.1.1.4.1

Factor $4$ out of $- 16$ .

$40 + 4 (- 4) (\frac{3 y}{4}) + 7 y = 30$

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

Step 2.2.1.1.4.2

Cancel the common factor.

$40 + 4 \cdot (- 4 \frac{3 y}{4}) + 7 y = 30$

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

Step 2.2.1.1.4.3

Rewrite the expression.

$40 - 4 (3 y) + 7 y = 30$

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

$40 - 4 (3 y) + 7 y = 30$

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

Step 2.2.1.1.5

Multiply $3$ by $- 4$ .

$40 - 12 y + 7 y = 30$

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

$40 - 12 y + 7 y = 30$

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

Step 2.2.1.2

Add $- 12 y$ and $7 y$ .

$40 - 5 y = 30$

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

$40 - 5 y = 30$

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

$40 - 5 y = 30$

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

$40 - 5 y = 30$

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

Step 3

Solve for $y$ in $40 - 5 y = 30$ .

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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 $40$ from both sides of the equation.

$- 5 y = 30 - 40$

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

Step 3.1.2

Subtract $40$ from $30$ .

$- 5 y = - 10$

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

$- 5 y = - 10$

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

Step 3.2

Divide each term in $- 5 y = - 10$ by $- 5$ and simplify.

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Step 3.2.1

Divide each term in $- 5 y = - 10$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{- 10}{- 5}$

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{- 10}{- 5}$

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

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 10}{- 5}$

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

$y = \frac{- 10}{- 5}$

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

$y = \frac{- 10}{- 5}$

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 10$ by $- 5$ .

$y = 2$

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

$y = 2$

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

$y = 2$

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

$y = 2$

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

Step 4

Replace all occurrences of $y$ with $2$ in each equation.

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Step 4.1

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

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

$y = 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{5}{2} + \frac{3 (2)}{4}$ .

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Step 4.2.1.1

Multiply $3$ by $2$ .

$x = - \frac{5}{2} + \frac{6}{4}$

$y = 2$

Step 4.2.1.2

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

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Step 4.2.1.2.1

Factor $2$ out of $6$ .

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

$y = 2$

Step 4.2.1.2.2

Cancel the common factors.

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Step 4.2.1.2.2.1

Factor $2$ out of $4$ .

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

$y = 2$

Step 4.2.1.2.2.2

Cancel the common factor.

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

$y = 2$

Step 4.2.1.2.2.3

Rewrite the expression.

$x = - \frac{5}{2} + \frac{3}{2}$

$y = 2$

$x = - \frac{5}{2} + \frac{3}{2}$

$y = 2$

$x = - \frac{5}{2} + \frac{3}{2}$

$y = 2$

Step 4.2.1.3

Combine the numerators over the common denominator.

$x = \frac{- 5 + 3}{2}$

$y = 2$

Step 4.2.1.4

Simplify the expression.

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Step 4.2.1.4.1

Add $- 5$ and $3$ .

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

$y = 2$

Step 4.2.1.4.2

Divide $- 2$ by $2$ .

$x = - 1$

$y = 2$

$x = - 1$

$y = 2$

$x = - 1$

$y = 2$

$x = - 1$

$y = 2$

$x = - 1$

$y = 2$

Step 5

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

$(- 1, 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 1, 2)$

Equation Form:

$x = - 1, y = 2$

Step 7