Linear Algebra Examples

$- 6 x + 28 y - 61 = 0$ , $8 x + 30 y - 29 = 0$

Step 1

Solve for $x$ in $- 6 x + 28 y - 61 = 0$ .

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Step 1.1

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

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Step 1.1.1

Subtract $28 y$ from both sides of the equation.

$- 6 x - 61 = - 28 y$

$8 x + 30 y - 29 = 0$

Step 1.1.2

Add $61$ to both sides of the equation.

$- 6 x = - 28 y + 61$

$8 x + 30 y - 29 = 0$

$- 6 x = - 28 y + 61$

$8 x + 30 y - 29 = 0$

Step 1.2

Divide each term in $- 6 x = - 28 y + 61$ by $- 6$ and simplify.

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Step 1.2.1

Divide each term in $- 6 x = - 28 y + 61$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 28 y}{- 6} + \frac{61}{- 6}$

$8 x + 30 y - 29 = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 6$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 28 y}{- 6} + \frac{61}{- 6}$

$8 x + 30 y - 29 = 0$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 28 y}{- 6} + \frac{61}{- 6}$

$8 x + 30 y - 29 = 0$

$x = \frac{- 28 y}{- 6} + \frac{61}{- 6}$

$8 x + 30 y - 29 = 0$

$x = \frac{- 28 y}{- 6} + \frac{61}{- 6}$

$8 x + 30 y - 29 = 0$

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 $- 28$ and $- 6$ .

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Step 1.2.3.1.1.1

Factor $- 2$ out of $- 28 y$ .

$x = \frac{- 2 (14 y)}{- 6} + \frac{61}{- 6}$

$8 x + 30 y - 29 = 0$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $- 2$ out of $- 6$ .

$x = \frac{- 2 (14 y)}{- 2 \cdot 3} + \frac{61}{- 6}$

$8 x + 30 y - 29 = 0$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{- 2 (14 y)}{- 2 \cdot 3} + \frac{61}{- 6}$

$8 x + 30 y - 29 = 0$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{14 y}{3} + \frac{61}{- 6}$

$8 x + 30 y - 29 = 0$

$x = \frac{14 y}{3} + \frac{61}{- 6}$

$8 x + 30 y - 29 = 0$

$x = \frac{14 y}{3} + \frac{61}{- 6}$

$8 x + 30 y - 29 = 0$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = \frac{14 y}{3} - \frac{61}{6}$

$8 x + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

$8 x + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

$8 x + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

$8 x + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

$8 x + 30 y - 29 = 0$

Step 2

Replace all occurrences of $x$ with $\frac{14 y}{3} - \frac{61}{6}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $8 x + 30 y - 29 = 0$ with $\frac{14 y}{3} - \frac{61}{6}$ .

$8 (\frac{14 y}{3} - \frac{61}{6}) + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $8 (\frac{14 y}{3} - \frac{61}{6}) + 30 y - 29$ .

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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.

$8 (\frac{14 y}{3}) + 8 (- \frac{61}{6}) + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.1.2

Multiply $8 \frac{14 y}{3}$ .

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Step 2.2.1.1.2.1

Combine $8$ and $\frac{14 y}{3}$ .

$\frac{8 (14 y)}{3} + 8 (- \frac{61}{6}) + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.1.2.2

Multiply $14$ by $8$ .

$\frac{112 y}{3} + 8 (- \frac{61}{6}) + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

$\frac{112 y}{3} + 8 (- \frac{61}{6}) + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

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 $- \frac{61}{6}$ into the numerator.

$\frac{112 y}{3} + 8 (\frac{- 61}{6}) + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.1.3.2

Factor $2$ out of $8$ .

$\frac{112 y}{3} + 2 (4) (\frac{- 61}{6}) + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.1.3.3

Factor $2$ out of $6$ .

$\frac{112 y}{3} + 2 \cdot (4 (\frac{- 61}{2 \cdot 3})) + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.1.3.4

Cancel the common factor.

$\frac{112 y}{3} + 2 \cdot (4 (\frac{- 61}{2 \cdot 3})) + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.1.3.5

Rewrite the expression.

$\frac{112 y}{3} + 4 (\frac{- 61}{3}) + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

$\frac{112 y}{3} + 4 (\frac{- 61}{3}) + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.1.4

Combine $4$ and $\frac{- 61}{3}$ .

$\frac{112 y}{3} + \frac{4 \cdot - 61}{3} + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.1.5

Multiply $4$ by $- 61$ .

$\frac{112 y}{3} + \frac{- 244}{3} + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.1.6

Move the negative in front of the fraction.

$\frac{112 y}{3} - \frac{244}{3} + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

$\frac{112 y}{3} - \frac{244}{3} + 30 y - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.2

To write $30 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{112 y}{3} + 30 y \cdot \frac{3}{3} - \frac{244}{3} - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.3

Combine $30 y$ and $\frac{3}{3}$ .

$\frac{112 y}{3} + \frac{30 y \cdot 3}{3} - \frac{244}{3} - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{112 y + 30 y \cdot 3}{3} - \frac{244}{3} - 29 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$- 29 + \frac{112 y + 30 y \cdot 3 - 244}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.6

Multiply $3$ by $30$ .

$- 29 + \frac{112 y + 90 y - 244}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.7

Add $112 y$ and $90 y$ .

$- 29 + \frac{202 y - 244}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.8

Factor $2$ out of $202 y - 244$ .

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Step 2.2.1.8.1

Factor $2$ out of $202 y$ .

$- 29 + \frac{2 (101 y) - 244}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.8.2

Factor $2$ out of $- 244$ .

$- 29 + \frac{2 (101 y) + 2 (- 122)}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.8.3

Factor $2$ out of $2 (101 y) + 2 (- 122)$ .

$- 29 + \frac{2 (101 y - 122)}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

$- 29 + \frac{2 (101 y - 122)}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.9

To write $- 29$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 29 \cdot \frac{3}{3} + \frac{2 (101 y - 122)}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.10

Simplify terms.

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Step 2.2.1.10.1

Combine $- 29$ and $\frac{3}{3}$ .

$\frac{- 29 \cdot 3}{3} + \frac{2 (101 y - 122)}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.10.2

Combine the numerators over the common denominator.

$\frac{- 29 \cdot 3 + 2 (101 y - 122)}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

$\frac{- 29 \cdot 3 + 2 (101 y - 122)}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.11

Simplify the numerator.

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Step 2.2.1.11.1

Multiply $- 29$ by $3$ .

$\frac{- 87 + 2 (101 y - 122)}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.11.2

Apply the distributive property.

$\frac{- 87 + 2 (101 y) + 2 \cdot - 122}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.11.3

Multiply $101$ by $2$ .

$\frac{- 87 + 202 y + 2 \cdot - 122}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.11.4

Multiply $2$ by $- 122$ .

$\frac{- 87 + 202 y - 244}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 2.2.1.11.5

Subtract $244$ from $- 87$ .

$\frac{202 y - 331}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

$\frac{202 y - 331}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

$\frac{202 y - 331}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

$\frac{202 y - 331}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

$\frac{202 y - 331}{3} = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 3

Solve for $y$ in $\frac{202 y - 331}{3} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$202 y - 331 = 0$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 3.2

Solve the equation for $y$ .

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Step 3.2.1

Add $331$ to both sides of the equation.

$202 y = 331$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 3.2.2

Divide each term in $202 y = 331$ by $202$ and simplify.

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Step 3.2.2.1

Divide each term in $202 y = 331$ by $202$ .

$\frac{202 y}{202} = \frac{331}{202}$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $202$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{202 y}{202} = \frac{331}{202}$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 3.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{331}{202}$

$x = \frac{14 y}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

$x = \frac{14 y}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

$x = \frac{14 y}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

$x = \frac{14 y}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

$x = \frac{14 y}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

$x = \frac{14 y}{3} - \frac{61}{6}$

Step 4

Replace all occurrences of $y$ with $\frac{331}{202}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{14 y}{3} - \frac{61}{6}$ with $\frac{331}{202}$ .

$x = \frac{14 (\frac{331}{202})}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{14 (\frac{331}{202})}{3} - \frac{61}{6}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Combine $14$ and $\frac{331}{202}$ .

$x = \frac{\frac{14 \cdot 331}{202}}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

Step 4.2.1.1.2

Multiply $14$ by $331$ .

$x = \frac{\frac{4634}{202}}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

Step 4.2.1.1.3

Cancel the common factor of $4634$ and $202$ .

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Step 4.2.1.1.3.1

Factor $2$ out of $4634$ .

$x = \frac{\frac{2 (2317)}{202}}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

Step 4.2.1.1.3.2

Cancel the common factors.

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Step 4.2.1.1.3.2.1

Factor $2$ out of $202$ .

$x = \frac{\frac{2 \cdot 2317}{2 \cdot 101}}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

Step 4.2.1.1.3.2.2

Cancel the common factor.

$x = \frac{\frac{2 \cdot 2317}{2 \cdot 101}}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

Step 4.2.1.1.3.2.3

Rewrite the expression.

$x = \frac{\frac{2317}{101}}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

$x = \frac{\frac{2317}{101}}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

$x = \frac{\frac{2317}{101}}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

Step 4.2.1.1.4

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{2317}{101} \cdot \frac{1}{3} - \frac{61}{6}$

$y = \frac{331}{202}$

Step 4.2.1.1.5

Multiply $\frac{2317}{101} \cdot \frac{1}{3}$ .

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Step 4.2.1.1.5.1

Multiply $\frac{2317}{101}$ by $\frac{1}{3}$ .

$x = \frac{2317}{101 \cdot 3} - \frac{61}{6}$

$y = \frac{331}{202}$

Step 4.2.1.1.5.2

Multiply $101$ by $3$ .

$x = \frac{2317}{303} - \frac{61}{6}$

$y = \frac{331}{202}$

$x = \frac{2317}{303} - \frac{61}{6}$

$y = \frac{331}{202}$

$x = \frac{2317}{303} - \frac{61}{6}$

$y = \frac{331}{202}$

Step 4.2.1.2

To write $\frac{2317}{303}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{2317}{303} \cdot \frac{2}{2} - \frac{61}{6}$

$y = \frac{331}{202}$

Step 4.2.1.3

To write $- \frac{61}{6}$ as a fraction with a common denominator, multiply by $\frac{101}{101}$ .

$x = \frac{2317}{303} \cdot \frac{2}{2} - \frac{61}{6} \cdot \frac{101}{101}$

$y = \frac{331}{202}$

Step 4.2.1.4

Write each expression with a common denominator of $606$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.4.1

Multiply $\frac{2317}{303}$ by $\frac{2}{2}$ .

$x = \frac{2317 \cdot 2}{303 \cdot 2} - \frac{61}{6} \cdot \frac{101}{101}$

$y = \frac{331}{202}$

Step 4.2.1.4.2

Multiply $303$ by $2$ .

$x = \frac{2317 \cdot 2}{606} - \frac{61}{6} \cdot \frac{101}{101}$

$y = \frac{331}{202}$

Step 4.2.1.4.3

Multiply $\frac{61}{6}$ by $\frac{101}{101}$ .

$x = \frac{2317 \cdot 2}{606} - \frac{61 \cdot 101}{6 \cdot 101}$

$y = \frac{331}{202}$

Step 4.2.1.4.4

Multiply $6$ by $101$ .

$x = \frac{2317 \cdot 2}{606} - \frac{61 \cdot 101}{606}$

$y = \frac{331}{202}$

$x = \frac{2317 \cdot 2}{606} - \frac{61 \cdot 101}{606}$

$y = \frac{331}{202}$

Step 4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{2317 \cdot 2 - 61 \cdot 101}{606}$

$y = \frac{331}{202}$

Step 4.2.1.6

Simplify the numerator.

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Step 4.2.1.6.1

Multiply $2317$ by $2$ .

$x = \frac{4634 - 61 \cdot 101}{606}$

$y = \frac{331}{202}$

Step 4.2.1.6.2

Multiply $- 61$ by $101$ .

$x = \frac{4634 - 6161}{606}$

$y = \frac{331}{202}$

Step 4.2.1.6.3

Subtract $6161$ from $4634$ .

$x = \frac{- 1527}{606}$

$y = \frac{331}{202}$

$x = \frac{- 1527}{606}$

$y = \frac{331}{202}$

Step 4.2.1.7

Cancel the common factor of $- 1527$ and $606$ .

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Step 4.2.1.7.1

Factor $3$ out of $- 1527$ .

$x = \frac{3 (- 509)}{606}$

$y = \frac{331}{202}$

Step 4.2.1.7.2

Cancel the common factors.

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Step 4.2.1.7.2.1

Factor $3$ out of $606$ .

$x = \frac{3 \cdot - 509}{3 \cdot 202}$

$y = \frac{331}{202}$

Step 4.2.1.7.2.2

Cancel the common factor.

$x = \frac{3 \cdot - 509}{3 \cdot 202}$

$y = \frac{331}{202}$

Step 4.2.1.7.2.3

Rewrite the expression.

$x = \frac{- 509}{202}$

$y = \frac{331}{202}$

$x = \frac{- 509}{202}$

$y = \frac{331}{202}$

$x = \frac{- 509}{202}$

$y = \frac{331}{202}$

Step 4.2.1.8

Move the negative in front of the fraction.

$x = - \frac{509}{202}$

$y = \frac{331}{202}$

$x = - \frac{509}{202}$

$y = \frac{331}{202}$

$x = - \frac{509}{202}$

$y = \frac{331}{202}$

$x = - \frac{509}{202}$

$y = \frac{331}{202}$

Step 5

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

$(- \frac{509}{202}, \frac{331}{202})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{509}{202}, \frac{331}{202})$

Equation Form:

$x = - \frac{509}{202}, y = \frac{331}{202}$

Step 7