Linear Algebra Examples

$4 x + 4 y = 1$ , $6 x - y = 1$

Solve for $x$ in the first equation.

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

$4 x = - 4 y + 1$

$6 x - y = 1$

Divide each term by $4$ and simplify.

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Divide each term in $4 x = - 4 y + 1$ by $4$ .

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

$6 x - y = 1$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

$6 x - y = 1$

Divide $x$ by $1$ .

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

$6 x - y = 1$

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

$6 x - y = 1$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

$6 x - y = 1$

Divide $y$ by $1$ .

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

$6 x - y = 1$

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

$6 x - y = 1$

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

$6 x - y = 1$

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

$6 x - y = 1$

Replace all occurrences of $x$ with the solution found by solving the last equation for $x$ . In this case, the value substituted is $- y + \frac{1}{4}$ .

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

$6 (- y + \frac{1}{4}) - y = 1$

Solve for $y$ in the second equation.

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Simplify the left side.

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Simplify each term.

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Apply the distributive property.

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

$6 (- y) + 6 (\frac{1}{4}) - y = 1$

Multiply $- 1$ by $6$ .

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

$- 6 y + 6 (\frac{1}{4}) - y = 1$

Cancel the common factor of $2$ .

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Write $6$ as a fraction with denominator $1$ .

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

$- 6 y + \frac{6}{1} \cdot \frac{1}{4} - y = 1$

Factor out the greatest common factor $2$ .

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

$- 6 y + \frac{2 \cdot 3}{1} \cdot \frac{1}{2 \cdot 2} - y = 1$

Cancel the common factor.

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

$- 6 y + \frac{2 \cdot 3}{1} \cdot \frac{1}{2 \cdot 2} - y = 1$

Rewrite the expression.

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

$- 6 y + \frac{3}{1} \cdot \frac{1}{2} - y = 1$

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

$- 6 y + \frac{3}{1} \cdot \frac{1}{2} - y = 1$

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

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

$- 6 y + \frac{3}{2} - y = 1$

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

$- 6 y + \frac{3}{2} - y = 1$

Subtract $y$ from $- 6 y$ .

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

$- 7 y + \frac{3}{2} = 1$

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

$- 7 y + \frac{3}{2} = 1$

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

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Subtract $\frac{3}{2}$ from both sides of the equation.

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

$- 7 y = - \frac{3}{2} + 1$

Simplify the right side of the equation.

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To write $\frac{1}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

$- 7 y = - \frac{3}{2} + \frac{1}{1} \cdot \frac{2}{2}$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

$- 7 y = - \frac{3}{2} + \frac{1 \cdot 2}{1 \cdot 2}$

Multiply $2$ by $1$ .

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

$- 7 y = - \frac{3}{2} + \frac{1 \cdot 2}{2}$

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

$- 7 y = - \frac{3}{2} + \frac{1 \cdot 2}{2}$

Combine the numerators over the common denominator.

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

$- 7 y = \frac{- 1 \cdot 3 + 1 \cdot 2}{2}$

Simplify the numerator.

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Multiply $- 1$ by $3$ .

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

$- 7 y = \frac{- 3 + 1 \cdot 2}{2}$

Multiply $1$ by $2$ .

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

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

Add $- 3$ and $2$ .

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

$- 7 y = \frac{- 1}{2}$

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

$- 7 y = \frac{- 1}{2}$

Move the negative in front of the fraction.

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

$- 7 y = - \frac{1}{2}$

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

$- 7 y = - \frac{1}{2}$

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

$- 7 y = - \frac{1}{2}$

Divide each term by $- 7$ and simplify.

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Divide each term in $- 7 y = - \frac{1}{2}$ by $- 7$ .

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

$- \frac{7 y}{- 7} = - \frac{1}{2} \cdot (- \frac{1}{7})$

Simplify the left side of the $e q u a t i o n$ by cancelling the common factors.

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Reduce the expression by cancelling the common factors.

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Factor $7$ out of $7 y$ .

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

$- \frac{7 (y)}{- 7} = - \frac{1}{2} \cdot (- \frac{1}{7})$

Move the negative one from the denominator of $\frac{y}{- 1}$ .

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

$- (- 1 \cdot y) = - \frac{1}{2} \cdot (- \frac{1}{7})$

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

$- (- 1 \cdot y) = - \frac{1}{2} \cdot (- \frac{1}{7})$

Simplify the expression.

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Multiply $- 1$ by $y$ .

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

$- (- 1 y) = - \frac{1}{2} \cdot (- \frac{1}{7})$

Rewrite $- 1 y$ as $- y$ .

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

$y = - \frac{1}{2} \cdot (- \frac{1}{7})$

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

$y = - \frac{1}{2} \cdot (- \frac{1}{7})$

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

$y = - \frac{1}{2} \cdot (- \frac{1}{7})$

Simplify the right side of the equation.

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Multiply $- \frac{1}{2}$ by $- \frac{1}{7}$ .

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

$y = - \frac{1}{2} \cdot (- \frac{1}{7})$

Simplify $- \frac{1}{2} (- \frac{1}{7})$ .

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Multiply $- 1$ by $- 1$ .

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

$y = 1 (\frac{1}{2}) \cdot \frac{1}{7}$

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

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

$y = \frac{1}{2} \cdot \frac{1}{7}$

Multiply $\frac{1}{2}$ and $\frac{1}{7}$ .

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

$y = \frac{1}{2 \cdot 7}$

Multiply $2$ by $7$ .

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

$y = \frac{1}{14}$

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

$y = \frac{1}{14}$

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

$y = \frac{1}{14}$

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

$y = \frac{1}{14}$

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

$y = \frac{1}{14}$

Replace all occurrences of $y$ with the solution found by solving the last equation for $y$ . In this case, the value substituted is $\frac{1}{14}$ .

$x = - (\frac{1}{14}) + \frac{1}{4}$

$y = \frac{1}{14}$

Simplify the right side.

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Multiply $- 1$ by $\frac{1}{14}$ .

$x = - \frac{1}{14} + \frac{1}{4}$

$y = \frac{1}{14}$

To write $\frac{1}{14}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = - \frac{1}{14} \cdot \frac{2}{2} + \frac{1}{4}$

$y = \frac{1}{14}$

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = - \frac{1}{14} \cdot \frac{2}{2} + \frac{1}{4} \cdot \frac{7}{7}$

$y = \frac{1}{14}$

Write each expression with a common denominator of $28$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = - \frac{1 \cdot 2}{14 \cdot 2} + \frac{1}{4} \cdot \frac{7}{7}$

$y = \frac{1}{14}$

Multiply $14$ by $2$ .

$x = - \frac{1 \cdot 2}{28} + \frac{1}{4} \cdot \frac{7}{7}$

$y = \frac{1}{14}$

Combine.

$x = - \frac{1 \cdot 2}{28} + \frac{1 \cdot 7}{4 \cdot 7}$

$y = \frac{1}{14}$

Multiply $4$ by $7$ .

$x = - \frac{1 \cdot 2}{28} + \frac{1 \cdot 7}{28}$

$y = \frac{1}{14}$

$x = - \frac{1 \cdot 2}{28} + \frac{1 \cdot 7}{28}$

$y = \frac{1}{14}$

Combine the numerators over the common denominator.

$x = \frac{- (1 \cdot 2) + 1 \cdot 7}{28}$

$y = \frac{1}{14}$

Simplify the numerator.

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Multiply $1$ by $2$ .

$x = \frac{- 1 \cdot 2 + 1 \cdot 7}{28}$

$y = \frac{1}{14}$

Multiply $- 1$ by $2$ .

$x = \frac{- 2 + 1 \cdot 7}{28}$

$y = \frac{1}{14}$

Multiply $1$ by $7$ .

$x = \frac{- 2 + 7}{28}$

$y = \frac{1}{14}$

Add $- 2$ and $7$ .

$x = \frac{5}{28}$

$y = \frac{1}{14}$

$x = \frac{5}{28}$

$y = \frac{1}{14}$

$x = \frac{5}{28}$

$y = \frac{1}{14}$

The solution can be shown in both point and equation forms.

Point Form:

$(\frac{5}{28}, \frac{1}{14})$

Equation Form: $x = \frac{5}{28} y = \frac{1}{14}$

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