Linear Algebra Examples

$4 x + 7 y = 15$ , $6 x - 5 y = 21$

Step 1

Solve for $x$ in $4 x + 7 y = 15$ .

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Step 1.1

Subtract $7 y$ from both sides of the equation.

$4 x = 15 - 7 y$

$6 x - 5 y = 21$

Step 1.2

Divide each term in $4 x = 15 - 7 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = 15 - 7 y$ by $4$ .

$\frac{4 x}{4} = \frac{15}{4} + \frac{- 7 y}{4}$

$6 x - 5 y = 21$

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.

$\frac{4 x}{4} = \frac{15}{4} + \frac{- 7 y}{4}$

$6 x - 5 y = 21$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{15}{4} + \frac{- 7 y}{4}$

$6 x - 5 y = 21$

$x = \frac{15}{4} + \frac{- 7 y}{4}$

$6 x - 5 y = 21$

$x = \frac{15}{4} + \frac{- 7 y}{4}$

$6 x - 5 y = 21$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x = \frac{15}{4} - \frac{7 y}{4}$

$6 x - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

$6 x - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

$6 x - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

$6 x - 5 y = 21$

Step 2

Replace all occurrences of $x$ with $\frac{15}{4} - \frac{7 y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $6 x - 5 y = 21$ with $\frac{15}{4} - \frac{7 y}{4}$ .

$6 (\frac{15}{4} - \frac{7 y}{4}) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $6 (\frac{15}{4} - \frac{7 y}{4}) - 5 y$ .

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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.

$6 (\frac{15}{4}) + 6 (- \frac{7 y}{4}) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Factor $2$ out of $6$ .

$2 (3) (\frac{15}{4}) + 6 (- \frac{7 y}{4}) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.2.2

Factor $2$ out of $4$ .

$2 \cdot (3 (\frac{15}{2 \cdot 2})) + 6 (- \frac{7 y}{4}) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.2.3

Cancel the common factor.

$2 \cdot (3 (\frac{15}{2 \cdot 2})) + 6 (- \frac{7 y}{4}) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.2.4

Rewrite the expression.

$3 (\frac{15}{2}) + 6 (- \frac{7 y}{4}) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

$3 (\frac{15}{2}) + 6 (- \frac{7 y}{4}) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3

Combine $3$ and $\frac{15}{2}$ .

$\frac{3 \cdot 15}{2} + 6 (- \frac{7 y}{4}) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.4

Multiply $3$ by $15$ .

$\frac{45}{2} + 6 (- \frac{7 y}{4}) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.5

Cancel the common factor of $2$ .

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Step 2.2.1.1.5.1

Move the leading negative in $- \frac{7 y}{4}$ into the numerator.

$\frac{45}{2} + 6 (\frac{- 7 y}{4}) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.5.2

Factor $2$ out of $6$ .

$\frac{45}{2} + 2 (3) (\frac{- 7 y}{4}) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.5.3

Factor $2$ out of $4$ .

$\frac{45}{2} + 2 \cdot (3 (\frac{- 7 y}{2 \cdot 2})) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.5.4

Cancel the common factor.

$\frac{45}{2} + 2 \cdot (3 (\frac{- 7 y}{2 \cdot 2})) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.5.5

Rewrite the expression.

$\frac{45}{2} + 3 (\frac{- 7 y}{2}) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

$\frac{45}{2} + 3 (\frac{- 7 y}{2}) - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.6

Combine $3$ and $\frac{- 7 y}{2}$ .

$\frac{45}{2} + \frac{3 (- 7 y)}{2} - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.7

Multiply $- 7$ by $3$ .

$\frac{45}{2} + \frac{- 21 y}{2} - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.1.8

Move the negative in front of the fraction.

$\frac{45}{2} - \frac{21 y}{2} - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

$\frac{45}{2} - \frac{21 y}{2} - 5 y = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.2

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

$\frac{45}{2} - \frac{21 y}{2} - 5 y \cdot \frac{2}{2} = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.3

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

$\frac{45}{2} - \frac{21 y}{2} + \frac{- 5 y \cdot 2}{2} = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{45}{2} + \frac{- 21 y - 5 y \cdot 2}{2} = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{45 - 21 y - 5 y \cdot 2}{2} = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.6

Multiply $2$ by $- 5$ .

$\frac{45 - 21 y - 10 y}{2} = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 2.2.1.7

Subtract $10 y$ from $- 21 y$ .

$\frac{45 - 31 y}{2} = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

$\frac{45 - 31 y}{2} = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

$\frac{45 - 31 y}{2} = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

$\frac{45 - 31 y}{2} = 21$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 3

Solve for $y$ in $\frac{45 - 31 y}{2} = 21$ .

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Step 3.1

Multiply both sides by $2$ .

$\frac{45 - 31 y}{2} \cdot 2 = 21 \cdot 2$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{45 - 31 y}{2} \cdot 2$ .

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Step 3.2.1.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{45 - 31 y}{2} \cdot 2 = 21 \cdot 2$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 3.2.1.1.1.2

Rewrite the expression.

$45 - 31 y = 21 \cdot 2$

$x = \frac{15}{4} - \frac{7 y}{4}$

$45 - 31 y = 21 \cdot 2$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 3.2.1.1.2

Reorder $45$ and $- 31 y$ .

$- 31 y + 45 = 21 \cdot 2$

$x = \frac{15}{4} - \frac{7 y}{4}$

$- 31 y + 45 = 21 \cdot 2$

$x = \frac{15}{4} - \frac{7 y}{4}$

$- 31 y + 45 = 21 \cdot 2$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $21$ by $2$ .

$- 31 y + 45 = 42$

$x = \frac{15}{4} - \frac{7 y}{4}$

$- 31 y + 45 = 42$

$x = \frac{15}{4} - \frac{7 y}{4}$

$- 31 y + 45 = 42$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $45$ from both sides of the equation.

$- 31 y = 42 - 45$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 3.3.1.2

Subtract $45$ from $42$ .

$- 31 y = - 3$

$x = \frac{15}{4} - \frac{7 y}{4}$

$- 31 y = - 3$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 3.3.2

Divide each term in $- 31 y = - 3$ by $- 31$ and simplify.

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Step 3.3.2.1

Divide each term in $- 31 y = - 3$ by $- 31$ .

$\frac{- 31 y}{- 31} = \frac{- 3}{- 31}$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $- 31$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{- 31 y}{- 31} = \frac{- 3}{- 31}$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3}{- 31}$

$x = \frac{15}{4} - \frac{7 y}{4}$

$y = \frac{- 3}{- 31}$

$x = \frac{15}{4} - \frac{7 y}{4}$

$y = \frac{- 3}{- 31}$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Dividing two negative values results in a positive value.

$y = \frac{3}{31}$

$x = \frac{15}{4} - \frac{7 y}{4}$

$y = \frac{3}{31}$

$x = \frac{15}{4} - \frac{7 y}{4}$

$y = \frac{3}{31}$

$x = \frac{15}{4} - \frac{7 y}{4}$

$y = \frac{3}{31}$

$x = \frac{15}{4} - \frac{7 y}{4}$

$y = \frac{3}{31}$

$x = \frac{15}{4} - \frac{7 y}{4}$

Step 4

Replace all occurrences of $y$ with $\frac{3}{31}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{15}{4} - \frac{7 y}{4}$ with $\frac{3}{31}$ .

$x = \frac{15}{4} - \frac{7 (\frac{3}{31})}{4}$

$y = \frac{3}{31}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{15}{4} - \frac{7 (\frac{3}{31})}{4}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{15 - 7 (\frac{3}{31})}{4}$

$y = \frac{3}{31}$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Multiply $- 7 (\frac{3}{31})$ .

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Step 4.2.1.2.1.1

Combine $- 7$ and $\frac{3}{31}$ .

$x = \frac{15 + \frac{- 7 \cdot 3}{31}}{4}$

$y = \frac{3}{31}$

Step 4.2.1.2.1.2

Multiply $- 7$ by $3$ .

$x = \frac{15 + \frac{- 21}{31}}{4}$

$y = \frac{3}{31}$

$x = \frac{15 + \frac{- 21}{31}}{4}$

$y = \frac{3}{31}$

Step 4.2.1.2.2

Move the negative in front of the fraction.

$x = \frac{15 - \frac{21}{31}}{4}$

$y = \frac{3}{31}$

$x = \frac{15 - \frac{21}{31}}{4}$

$y = \frac{3}{31}$

Step 4.2.1.3

To write $15$ as a fraction with a common denominator, multiply by $\frac{31}{31}$ .

$x = \frac{15 \cdot \frac{31}{31} - \frac{21}{31}}{4}$

$y = \frac{3}{31}$

Step 4.2.1.4

Combine $15$ and $\frac{31}{31}$ .

$x = \frac{\frac{15 \cdot 31}{31} - \frac{21}{31}}{4}$

$y = \frac{3}{31}$

Step 4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{15 \cdot 31 - 21}{31}}{4}$

$y = \frac{3}{31}$

Step 4.2.1.6

Simplify the numerator.

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Step 4.2.1.6.1

Multiply $15$ by $31$ .

$x = \frac{\frac{465 - 21}{31}}{4}$

$y = \frac{3}{31}$

Step 4.2.1.6.2

Subtract $21$ from $465$ .

$x = \frac{\frac{444}{31}}{4}$

$y = \frac{3}{31}$

$x = \frac{\frac{444}{31}}{4}$

$y = \frac{3}{31}$

Step 4.2.1.7

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{444}{31} \cdot \frac{1}{4}$

$y = \frac{3}{31}$

Step 4.2.1.8

Cancel the common factor of $4$ .

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Step 4.2.1.8.1

Factor $4$ out of $444$ .

$x = \frac{4 (111)}{31} \cdot \frac{1}{4}$

$y = \frac{3}{31}$

Step 4.2.1.8.2

Cancel the common factor.

$x = \frac{4 \cdot 111}{31} \cdot \frac{1}{4}$

$y = \frac{3}{31}$

Step 4.2.1.8.3

Rewrite the expression.

$x = \frac{111}{31}$

$y = \frac{3}{31}$

$x = \frac{111}{31}$

$y = \frac{3}{31}$

$x = \frac{111}{31}$

$y = \frac{3}{31}$

$x = \frac{111}{31}$

$y = \frac{3}{31}$

$x = \frac{111}{31}$

$y = \frac{3}{31}$

Step 5

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

$(\frac{111}{31}, \frac{3}{31})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{111}{31}, \frac{3}{31})$

Equation Form:

$x = \frac{111}{31}, y = \frac{3}{31}$

Step 7