Linear Algebra Examples

$4 x + 4 y = 5$ , $2 x - 8 y = - 5$

Step 1

Solve for $x$ in $4 x + 4 y = 5$ .

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Step 1.1

Subtract $4 y$ from both sides of the equation.

$4 x = 5 - 4 y$

$2 x - 8 y = - 5$

Step 1.2

Divide each term in $4 x = 5 - 4 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = 5 - 4 y$ by $4$ .

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

$2 x - 8 y = - 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$2 x - 8 y = - 5$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$2 x - 8 y = - 5$

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

$2 x - 8 y = - 5$

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

$2 x - 8 y = - 5$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

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

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Step 1.2.3.1.1

Factor $4$ out of $- 4 y$ .

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

$2 x - 8 y = - 5$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $4$ out of $4$ .

$x = \frac{5}{4} + \frac{4 (- y)}{4 (1)}$

$2 x - 8 y = - 5$

Step 1.2.3.1.2.2

Cancel the common factor.

$x = \frac{5}{4} + \frac{4 (- y)}{4 \cdot 1}$

$2 x - 8 y = - 5$

Step 1.2.3.1.2.3

Rewrite the expression.

$x = \frac{5}{4} + \frac{- y}{1}$

$2 x - 8 y = - 5$

Step 1.2.3.1.2.4

Divide $- y$ by $1$ .

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

$2 x - 8 y = - 5$

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

$2 x - 8 y = - 5$

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

$2 x - 8 y = - 5$

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

$2 x - 8 y = - 5$

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

$2 x - 8 y = - 5$

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

$2 x - 8 y = - 5$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $2 x - 8 y = - 5$ with $\frac{5}{4} - y$ .

$2 (\frac{5}{4} - y) - 8 y = - 5$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (\frac{5}{4} - y) - 8 y$ .

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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.

$2 (\frac{5}{4}) + 2 (- y) - 8 y = - 5$

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Factor $2$ out of $4$ .

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

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

Step 2.2.1.1.2.2

Cancel the common factor.

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

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

Step 2.2.1.1.2.3

Rewrite the expression.

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

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

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

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

Step 2.2.1.1.3

Multiply $- 1$ by $2$ .

$\frac{5}{2} - 2 y - 8 y = - 5$

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

$\frac{5}{2} - 2 y - 8 y = - 5$

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

Step 2.2.1.2

Subtract $8 y$ from $- 2 y$ .

$\frac{5}{2} - 10 y = - 5$

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

$\frac{5}{2} - 10 y = - 5$

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

$\frac{5}{2} - 10 y = - 5$

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

$\frac{5}{2} - 10 y = - 5$

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

Step 3

Solve for $y$ in $\frac{5}{2} - 10 y = - 5$ .

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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 $\frac{5}{2}$ from both sides of the equation.

$- 10 y = - 5 - \frac{5}{2}$

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

Step 3.1.2

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

$- 10 y = - 5 \cdot \frac{2}{2} - \frac{5}{2}$

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

Step 3.1.3

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

$- 10 y = \frac{- 5 \cdot 2}{2} - \frac{5}{2}$

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

Step 3.1.4

Combine the numerators over the common denominator.

$- 10 y = \frac{- 5 \cdot 2 - 5}{2}$

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

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $- 5$ by $2$ .

$- 10 y = \frac{- 10 - 5}{2}$

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

Step 3.1.5.2

Subtract $5$ from $- 10$ .

$- 10 y = \frac{- 15}{2}$

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

$- 10 y = \frac{- 15}{2}$

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

Step 3.1.6

Move the negative in front of the fraction.

$- 10 y = - \frac{15}{2}$

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

$- 10 y = - \frac{15}{2}$

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

Step 3.2

Divide each term in $- 10 y = - \frac{15}{2}$ by $- 10$ and simplify.

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Step 3.2.1

Divide each term in $- 10 y = - \frac{15}{2}$ by $- 10$ .

$\frac{- 10 y}{- 10} = \frac{- \frac{15}{2}}{- 10}$

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 10$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 10 y}{- 10} = \frac{- \frac{15}{2}}{- 10}$

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

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{15}{2}}{- 10}$

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

$y = \frac{- \frac{15}{2}}{- 10}$

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

$y = \frac{- \frac{15}{2}}{- 10}$

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{15}{2} \cdot \frac{1}{- 10}$

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

Step 3.2.3.2

Cancel the common factor of $5$ .

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Step 3.2.3.2.1

Move the leading negative in $- \frac{15}{2}$ into the numerator.

$y = \frac{- 15}{2} \cdot \frac{1}{- 10}$

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

Step 3.2.3.2.2

Factor $5$ out of $- 15$ .

$y = \frac{5 (- 3)}{2} \cdot \frac{1}{- 10}$

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

Step 3.2.3.2.3

Factor $5$ out of $- 10$ .

$y = \frac{5 \cdot - 3}{2} \cdot \frac{1}{5 \cdot - 2}$

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

Step 3.2.3.2.4

Cancel the common factor.

$y = \frac{5 \cdot - 3}{2} \cdot \frac{1}{5 \cdot - 2}$

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

Step 3.2.3.2.5

Rewrite the expression.

$y = \frac{- 3}{2} \cdot \frac{1}{- 2}$

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

$y = \frac{- 3}{2} \cdot \frac{1}{- 2}$

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

Step 3.2.3.3

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

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

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

Step 3.2.3.4

Multiply $2$ by $- 2$ .

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

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

Step 3.2.3.5

Dividing two negative values results in a positive value.

$y = \frac{3}{4}$

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

$y = \frac{3}{4}$

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

$y = \frac{3}{4}$

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

$y = \frac{3}{4}$

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

Step 4

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

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Step 4.1

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

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

$y = \frac{3}{4}$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Combine the numerators over the common denominator.

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

$y = \frac{3}{4}$

Step 4.2.1.2

Subtract $3$ from $5$ .

$x = \frac{2}{4}$

$y = \frac{3}{4}$

Step 4.2.1.3

Cancel the common factor of $2$ and $4$ .

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Step 4.2.1.3.1

Factor $2$ out of $2$ .

$x = \frac{2 (1)}{4}$

$y = \frac{3}{4}$

Step 4.2.1.3.2

Cancel the common factors.

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Step 4.2.1.3.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 1}{2 \cdot 2}$

$y = \frac{3}{4}$

Step 4.2.1.3.2.2

Cancel the common factor.

$x = \frac{2 \cdot 1}{2 \cdot 2}$

$y = \frac{3}{4}$

Step 4.2.1.3.2.3

Rewrite the expression.

$x = \frac{1}{2}$

$y = \frac{3}{4}$

$x = \frac{1}{2}$

$y = \frac{3}{4}$

$x = \frac{1}{2}$

$y = \frac{3}{4}$

$x = \frac{1}{2}$

$y = \frac{3}{4}$

$x = \frac{1}{2}$

$y = \frac{3}{4}$

$x = \frac{1}{2}$

$y = \frac{3}{4}$

Step 5

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

$(\frac{1}{2}, \frac{3}{4})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{1}{2}, \frac{3}{4})$

Equation Form:

$x = \frac{1}{2}, y = \frac{3}{4}$

Step 7