Linear Algebra Examples

$\frac{1}{4} x - \frac{1}{5} y = 9$ , $5 x - y = 0$

Step 1

Solve for $x$ in $5 x - y = 0$ .

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Step 1.1

Add $y$ to both sides of the equation.

$5 x = y$

$\frac{1}{4} x - \frac{1}{5} y = 9$

Step 1.2

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

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Step 1.2.1

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

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

$\frac{1}{4} x - \frac{1}{5} y = 9$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$\frac{1}{4} x - \frac{1}{5} y = 9$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$\frac{1}{4} x - \frac{1}{5} y = 9$

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

$\frac{1}{4} x - \frac{1}{5} y = 9$

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

$\frac{1}{4} x - \frac{1}{5} y = 9$

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

$\frac{1}{4} x - \frac{1}{5} y = 9$

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

$\frac{1}{4} x - \frac{1}{5} y = 9$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $\frac{1}{4} x - \frac{1}{5} y = 9$ with $\frac{y}{5}$ .

$\frac{1}{4} \cdot (\frac{y}{5}) - \frac{1}{5} y = 9$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{1}{4} \cdot (\frac{y}{5}) - \frac{1}{5} y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Multiply $\frac{1}{4} (\frac{y}{5})$ .

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Step 2.2.1.1.1.1

Multiply $\frac{1}{4}$ by $\frac{y}{5}$ .

$\frac{y}{4 \cdot 5} - \frac{1}{5} y = 9$

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

Step 2.2.1.1.1.2

Multiply $4$ by $5$ .

$\frac{y}{20} - \frac{1}{5} y = 9$

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

$\frac{y}{20} - \frac{1}{5} y = 9$

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

Step 2.2.1.1.2

Combine $y$ and $\frac{1}{5}$ .

$\frac{y}{20} - \frac{y}{5} = 9$

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

$\frac{y}{20} - \frac{y}{5} = 9$

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

Step 2.2.1.2

To write $- \frac{y}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{y}{20} - \frac{y}{5} \cdot \frac{4}{4} = 9$

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

Step 2.2.1.3

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.3.1

Multiply $\frac{y}{5}$ by $\frac{4}{4}$ .

$\frac{y}{20} - \frac{y \cdot 4}{5 \cdot 4} = 9$

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

Step 2.2.1.3.2

Multiply $5$ by $4$ .

$\frac{y}{20} - \frac{y \cdot 4}{20} = 9$

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

$\frac{y}{20} - \frac{y \cdot 4}{20} = 9$

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

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{y - y \cdot 4}{20} = 9$

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

Step 2.2.1.5

Simplify the numerator.

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Step 2.2.1.5.1

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

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Step 2.2.1.5.1.1

Raise $y$ to the power of $1$ .

$\frac{y - y \cdot 4}{20} = 9$

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

Step 2.2.1.5.1.2

Factor $y$ out of $y^{1}$ .

$\frac{y \cdot 1 - y \cdot 4}{20} = 9$

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

Step 2.2.1.5.1.3

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

$\frac{y \cdot 1 + y (- 1 \cdot 4)}{20} = 9$

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

Step 2.2.1.5.1.4

Factor $y$ out of $y \cdot 1 + y (- 1 \cdot 4)$ .

$\frac{y (1 - 1 \cdot 4)}{20} = 9$

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

$\frac{y (1 - 1 \cdot 4)}{20} = 9$

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

Step 2.2.1.5.2

Multiply $- 1$ by $4$ .

$\frac{y (1 - 4)}{20} = 9$

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

Step 2.2.1.5.3

Subtract $4$ from $1$ .

$\frac{y \cdot - 3}{20} = 9$

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

$\frac{y \cdot - 3}{20} = 9$

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

Step 2.2.1.6

Simplify the expression.

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Step 2.2.1.6.1

Move $- 3$ to the left of $y$ .

$\frac{- 3 \cdot y}{20} = 9$

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

Step 2.2.1.6.2

Move the negative in front of the fraction.

$- \frac{3 y}{20} = 9$

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

$- \frac{3 y}{20} = 9$

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

$- \frac{3 y}{20} = 9$

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

$- \frac{3 y}{20} = 9$

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

$- \frac{3 y}{20} = 9$

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

Step 3

Solve for $y$ in $- \frac{3 y}{20} = 9$ .

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Step 3.1

Multiply both sides of the equation by $- \frac{20}{3}$ .

$- \frac{20}{3} \cdot (- \frac{3 y}{20}) = - \frac{20}{3} \cdot 9$

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $- \frac{20}{3} (- \frac{3 y}{20})$ .

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Step 3.2.1.1.1

Cancel the common factor of $20$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{20}{3}$ into the numerator.

$\frac{- 20}{3} \cdot (- \frac{3 y}{20}) = - \frac{20}{3} \cdot 9$

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

Step 3.2.1.1.1.2

Move the leading negative in $- \frac{3 y}{20}$ into the numerator.

$\frac{- 20}{3} \cdot \frac{- 3 y}{20} = - \frac{20}{3} \cdot 9$

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

Step 3.2.1.1.1.3

Factor $20$ out of $- 20$ .

$\frac{20 (- 1)}{3} \cdot \frac{- 3 y}{20} = - \frac{20}{3} \cdot 9$

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

Step 3.2.1.1.1.4

Cancel the common factor.

$\frac{20 \cdot - 1}{3} \cdot \frac{- 3 y}{20} = - \frac{20}{3} \cdot 9$

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

Step 3.2.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} \cdot (- 3 y) = - \frac{20}{3} \cdot 9$

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

$\frac{- 1}{3} \cdot (- 3 y) = - \frac{20}{3} \cdot 9$

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

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Factor $3$ out of $- 3 y$ .

$\frac{- 1}{3} \cdot (3 (- y)) = - \frac{20}{3} \cdot 9$

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

Step 3.2.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} \cdot (3 (- y)) = - \frac{20}{3} \cdot 9$

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

Step 3.2.1.1.2.3

Rewrite the expression.

$y = - \frac{20}{3} \cdot 9$

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

$y = - \frac{20}{3} \cdot 9$

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

Step 3.2.1.1.3

Multiply.

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Step 3.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{20}{3} \cdot 9$

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

Step 3.2.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{20}{3} \cdot 9$

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

$y = - \frac{20}{3} \cdot 9$

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

$y = - \frac{20}{3} \cdot 9$

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

$y = - \frac{20}{3} \cdot 9$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- \frac{20}{3} \cdot 9$ .

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Step 3.2.2.1.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1.1

Move the leading negative in $- \frac{20}{3}$ into the numerator.

$y = \frac{- 20}{3} \cdot 9$

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

Step 3.2.2.1.1.2

Factor $3$ out of $9$ .

$y = \frac{- 20}{3} \cdot (3 (3))$

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

Step 3.2.2.1.1.3

Cancel the common factor.

$y = \frac{- 20}{3} \cdot (3 \cdot 3)$

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

Step 3.2.2.1.1.4

Rewrite the expression.

$y = - 20 \cdot 3$

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

$y = - 20 \cdot 3$

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

Step 3.2.2.1.2

Multiply $- 20$ by $3$ .

$y = - 60$

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

$y = - 60$

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

$y = - 60$

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

$y = - 60$

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

$y = - 60$

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

Step 4

Replace all occurrences of $y$ with $- 60$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{y}{5}$ with $- 60$ .

$x = \frac{- 60}{5}$

$y = - 60$

Step 4.2

Simplify the right side.

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Step 4.2.1

Divide $- 60$ by $5$ .

$x = - 12$

$y = - 60$

$x = - 12$

$y = - 60$

$x = - 12$

$y = - 60$

Step 5

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

$(- 12, - 60)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 12, - 60)$

Equation Form:

$x = - 12, y = - 60$

Step 7