Linear Algebra Examples

$5 x + 3 y = - 7$ , $7 x - 3 y = 55$

Step 1

Solve for $x$ in $5 x + 3 y = - 7$ .

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Step 1.1

Subtract $3 y$ from both sides of the equation.

$5 x = - 7 - 3 y$

$7 x - 3 y = 55$

Step 1.2

Divide each term in $5 x = - 7 - 3 y$ by $5$ and simplify.

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Step 1.2.1

Divide each term in $5 x = - 7 - 3 y$ by $5$ .

$\frac{5 x}{5} = \frac{- 7}{5} + \frac{- 3 y}{5}$

$7 x - 3 y = 55$

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.

$\frac{5 x}{5} = \frac{- 7}{5} + \frac{- 3 y}{5}$

$7 x - 3 y = 55$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{5} + \frac{- 3 y}{5}$

$7 x - 3 y = 55$

$x = \frac{- 7}{5} + \frac{- 3 y}{5}$

$7 x - 3 y = 55$

$x = \frac{- 7}{5} + \frac{- 3 y}{5}$

$7 x - 3 y = 55$

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

Move the negative in front of the fraction.

$x = - \frac{7}{5} + \frac{- 3 y}{5}$

$7 x - 3 y = 55$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{7}{5} - \frac{3 y}{5}$

$7 x - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$7 x - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$7 x - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$7 x - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$7 x - 3 y = 55$

Step 2

Replace all occurrences of $x$ with $- \frac{7}{5} - \frac{3 y}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $7 x - 3 y = 55$ with $- \frac{7}{5} - \frac{3 y}{5}$ .

$7 (- \frac{7}{5} - \frac{3 y}{5}) - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $7 (- \frac{7}{5} - \frac{3 y}{5}) - 3 y$ .

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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.

$7 (- \frac{7}{5}) + 7 (- \frac{3 y}{5}) - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.1.2

Multiply $7 (- \frac{7}{5})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $7$ .

$- 7 (\frac{7}{5}) + 7 (- \frac{3 y}{5}) - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.1.2.2

Combine $- 7$ and $\frac{7}{5}$ .

$\frac{- 7 \cdot 7}{5} + 7 (- \frac{3 y}{5}) - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.1.2.3

Multiply $- 7$ by $7$ .

$\frac{- 49}{5} + 7 (- \frac{3 y}{5}) - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$\frac{- 49}{5} + 7 (- \frac{3 y}{5}) - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.1.3

Multiply $7 (- \frac{3 y}{5})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $7$ .

$\frac{- 49}{5} - 7 \frac{3 y}{5} - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.1.3.2

Combine $- 7$ and $\frac{3 y}{5}$ .

$\frac{- 49}{5} + \frac{- 7 (3 y)}{5} - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.1.3.3

Multiply $3$ by $- 7$ .

$\frac{- 49}{5} + \frac{- 21 y}{5} - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$\frac{- 49}{5} + \frac{- 21 y}{5} - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.1.4

Simplify each term.

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Step 2.2.1.1.4.1

Move the negative in front of the fraction.

$- \frac{49}{5} + \frac{- 21 y}{5} - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{49}{5} - \frac{21 y}{5} - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$- \frac{49}{5} - \frac{21 y}{5} - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$- \frac{49}{5} - \frac{21 y}{5} - 3 y = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.2

To write $- 3 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{49}{5} - \frac{21 y}{5} - 3 y \cdot \frac{5}{5} = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.3

Combine $- 3 y$ and $\frac{5}{5}$ .

$- \frac{49}{5} - \frac{21 y}{5} + \frac{- 3 y \cdot 5}{5} = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{49}{5} + \frac{- 21 y - 3 y \cdot 5}{5} = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 49 - 21 y - 3 y \cdot 5}{5} = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.6

Multiply $5$ by $- 3$ .

$\frac{- 49 - 21 y - 15 y}{5} = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.7

Subtract $15 y$ from $- 21 y$ .

$\frac{- 49 - 36 y}{5} = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.8

Rewrite $- 49$ as $- 1 (49)$ .

$\frac{- 1 \cdot 49 - 36 y}{5} = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.9

Factor $- 1$ out of $- 36 y$ .

$\frac{- 1 \cdot 49 - (36 y)}{5} = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.10

Factor $- 1$ out of $- 1 (49) - (36 y)$ .

$\frac{- 1 (49 + 36 y)}{5} = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 2.2.1.11

Move the negative in front of the fraction.

$- \frac{49 + 36 y}{5} = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$- \frac{49 + 36 y}{5} = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$- \frac{49 + 36 y}{5} = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$- \frac{49 + 36 y}{5} = 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 3

Solve for $y$ in $- \frac{49 + 36 y}{5} = 55$ .

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Step 3.1

Multiply both sides of the equation by $- 5$ .

$- 5 (- \frac{49 + 36 y}{5}) = - 5 \cdot 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

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 $- 5 (- \frac{49 + 36 y}{5})$ .

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Step 3.2.1.1.1

Cancel the common factor of $5$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{49 + 36 y}{5}$ into the numerator.

$- 5 \frac{- (49 + 36 y)}{5} = - 5 \cdot 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 3.2.1.1.1.2

Factor $5$ out of $- 5$ .

$5 (- 1) (\frac{- (49 + 36 y)}{5}) = - 5 \cdot 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 3.2.1.1.1.3

Cancel the common factor.

$5 \cdot (- 1 \frac{- (49 + 36 y)}{5}) = - 5 \cdot 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 3.2.1.1.1.4

Rewrite the expression.

$49 + 36 y = - 5 \cdot 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$49 + 36 y = - 5 \cdot 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (49 + 36 y) = - 5 \cdot 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 3.2.1.1.2.2

Multiply $49 + 36 y$ by $1$ .

$49 + 36 y = - 5 \cdot 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$49 + 36 y = - 5 \cdot 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$49 + 36 y = - 5 \cdot 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$49 + 36 y = - 5 \cdot 55$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 5$ by $55$ .

$49 + 36 y = - 275$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$49 + 36 y = - 275$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$49 + 36 y = - 275$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 3.3

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

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Step 3.3.1

Subtract $49$ from both sides of the equation.

$36 y = - 275 - 49$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 3.3.2

Subtract $49$ from $- 275$ .

$36 y = - 324$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$36 y = - 324$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 3.4

Divide each term in $36 y = - 324$ by $36$ and simplify.

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Step 3.4.1

Divide each term in $36 y = - 324$ by $36$ .

$\frac{36 y}{36} = \frac{- 324}{36}$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $36$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{36 y}{36} = \frac{- 324}{36}$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 324}{36}$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$y = \frac{- 324}{36}$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$y = \frac{- 324}{36}$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $- 324$ by $36$ .

$y = - 9$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$y = - 9$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$y = - 9$

$x = - \frac{7}{5} - \frac{3 y}{5}$

$y = - 9$

$x = - \frac{7}{5} - \frac{3 y}{5}$

Step 4

Replace all occurrences of $y$ with $- 9$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{7}{5} - \frac{3 y}{5}$ with $- 9$ .

$x = - \frac{7}{5} - \frac{3 (- 9)}{5}$

$y = - 9$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{7}{5} - \frac{3 (- 9)}{5}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 7 - 3 \cdot - 9}{5}$

$y = - 9$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $- 3$ by $- 9$ .

$x = \frac{- 7 + 27}{5}$

$y = - 9$

Step 4.2.1.2.2

Add $- 7$ and $27$ .

$x = \frac{20}{5}$

$y = - 9$

Step 4.2.1.2.3

Divide $20$ by $5$ .

$x = 4$

$y = - 9$

$x = 4$

$y = - 9$

$x = 4$

$y = - 9$

$x = 4$

$y = - 9$

$x = 4$

$y = - 9$

Step 5

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

$(4, - 9)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(4, - 9)$

Equation Form:

$x = 4, y = - 9$

Step 7