Linear Algebra Examples

$4 x - 5 y = - 17$ , $3 x + 2 y = - 7$

Step 1

Solve for $x$ in $4 x - 5 y = - 17$ .

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Step 1.1

Add $5 y$ to both sides of the equation.

$4 x = - 17 + 5 y$

$3 x + 2 y = - 7$

Step 1.2

Divide each term in $4 x = - 17 + 5 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = - 17 + 5 y$ by $4$ .

$\frac{4 x}{4} = \frac{- 17}{4} + \frac{5 y}{4}$

$3 x + 2 y = - 7$

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.

$\frac{4 x}{4} = \frac{- 17}{4} + \frac{5 y}{4}$

$3 x + 2 y = - 7$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 17}{4} + \frac{5 y}{4}$

$3 x + 2 y = - 7$

$x = \frac{- 17}{4} + \frac{5 y}{4}$

$3 x + 2 y = - 7$

$x = \frac{- 17}{4} + \frac{5 y}{4}$

$3 x + 2 y = - 7$

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.

$x = - \frac{17}{4} + \frac{5 y}{4}$

$3 x + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$3 x + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$3 x + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$3 x + 2 y = - 7$

Step 2

Replace all occurrences of $x$ with $- \frac{17}{4} + \frac{5 y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x + 2 y = - 7$ with $- \frac{17}{4} + \frac{5 y}{4}$ .

$3 (- \frac{17}{4} + \frac{5 y}{4}) + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (- \frac{17}{4} + \frac{5 y}{4}) + 2 y$ .

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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.

$3 (- \frac{17}{4}) + 3 (\frac{5 y}{4}) + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.1.2

Multiply $3 (- \frac{17}{4})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{17}{4}) + 3 (\frac{5 y}{4}) + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.1.2.2

Combine $- 3$ and $\frac{17}{4}$ .

$\frac{- 3 \cdot 17}{4} + 3 (\frac{5 y}{4}) + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.1.2.3

Multiply $- 3$ by $17$ .

$\frac{- 51}{4} + 3 (\frac{5 y}{4}) + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$\frac{- 51}{4} + 3 (\frac{5 y}{4}) + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.1.3

Multiply $3 \frac{5 y}{4}$ .

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Step 2.2.1.1.3.1

Combine $3$ and $\frac{5 y}{4}$ .

$\frac{- 51}{4} + \frac{3 (5 y)}{4} + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.1.3.2

Multiply $5$ by $3$ .

$\frac{- 51}{4} + \frac{15 y}{4} + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$\frac{- 51}{4} + \frac{15 y}{4} + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.1.4

Move the negative in front of the fraction.

$- \frac{51}{4} + \frac{15 y}{4} + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$- \frac{51}{4} + \frac{15 y}{4} + 2 y = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.2

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{51}{4} + \frac{15 y}{4} + 2 y \cdot \frac{4}{4} = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.3

Combine $2 y$ and $\frac{4}{4}$ .

$- \frac{51}{4} + \frac{15 y}{4} + \frac{2 y \cdot 4}{4} = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{51}{4} + \frac{15 y + 2 y \cdot 4}{4} = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 51 + 15 y + 2 y \cdot 4}{4} = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.6

Multiply $4$ by $2$ .

$\frac{- 51 + 15 y + 8 y}{4} = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.7

Add $15 y$ and $8 y$ .

$\frac{- 51 + 23 y}{4} = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.8

Rewrite $- 51$ as $- 1 (51)$ .

$\frac{- 1 \cdot 51 + 23 y}{4} = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.9

Factor $- 1$ out of $23 y$ .

$\frac{- 1 \cdot 51 - (- 23 y)}{4} = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.10

Factor $- 1$ out of $- 1 (51) - (- 23 y)$ .

$\frac{- 1 (51 - 23 y)}{4} = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 2.2.1.11

Move the negative in front of the fraction.

$- \frac{51 - 23 y}{4} = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$- \frac{51 - 23 y}{4} = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$- \frac{51 - 23 y}{4} = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$- \frac{51 - 23 y}{4} = - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 3

Solve for $y$ in $- \frac{51 - 23 y}{4} = - 7$ .

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Step 3.1

Multiply both sides of the equation by $- 4$ .

$- 4 (- \frac{51 - 23 y}{4}) = - 4 \cdot - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

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 $- 4 (- \frac{51 - 23 y}{4})$ .

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Step 3.2.1.1.1

Cancel the common factor of $4$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{51 - 23 y}{4}$ into the numerator.

$- 4 \frac{- (51 - 23 y)}{4} = - 4 \cdot - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 3.2.1.1.1.2

Factor $4$ out of $- 4$ .

$4 (- 1) (\frac{- (51 - 23 y)}{4}) = - 4 \cdot - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 3.2.1.1.1.3

Cancel the common factor.

$4 \cdot (- 1 \frac{- (51 - 23 y)}{4}) = - 4 \cdot - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 3.2.1.1.1.4

Rewrite the expression.

$51 - 23 y = - 4 \cdot - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$51 - 23 y = - 4 \cdot - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (51 - 23 y) = - 4 \cdot - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 3.2.1.1.2.2

Multiply $51 - 23 y$ by $1$ .

$51 - 23 y = - 4 \cdot - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$51 - 23 y = - 4 \cdot - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$51 - 23 y = - 4 \cdot - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$51 - 23 y = - 4 \cdot - 7$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 4$ by $- 7$ .

$51 - 23 y = 28$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$51 - 23 y = 28$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$51 - 23 y = 28$

$x = - \frac{17}{4} + \frac{5 y}{4}$

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 $51$ from both sides of the equation.

$- 23 y = 28 - 51$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 3.3.2

Subtract $51$ from $28$ .

$- 23 y = - 23$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$- 23 y = - 23$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 3.4

Divide each term in $- 23 y = - 23$ by $- 23$ and simplify.

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Step 3.4.1

Divide each term in $- 23 y = - 23$ by $- 23$ .

$\frac{- 23 y}{- 23} = \frac{- 23}{- 23}$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $- 23$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 23 y}{- 23} = \frac{- 23}{- 23}$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 23}{- 23}$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$y = \frac{- 23}{- 23}$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$y = \frac{- 23}{- 23}$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $- 23$ by $- 23$ .

$y = 1$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$y = 1$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$y = 1$

$x = - \frac{17}{4} + \frac{5 y}{4}$

$y = 1$

$x = - \frac{17}{4} + \frac{5 y}{4}$

Step 4

Replace all occurrences of $y$ with $1$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{17}{4} + \frac{5 y}{4}$ with $1$ .

$x = - \frac{17}{4} + \frac{5 (1)}{4}$

$y = 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{17}{4} + \frac{5 (1)}{4}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 17 + 5 (1)}{4}$

$y = 1$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $5$ by $1$ .

$x = \frac{- 17 + 5}{4}$

$y = 1$

Step 4.2.1.2.2

Add $- 17$ and $5$ .

$x = \frac{- 12}{4}$

$y = 1$

Step 4.2.1.2.3

Divide $- 12$ by $4$ .

$x = - 3$

$y = 1$

$x = - 3$

$y = 1$

$x = - 3$

$y = 1$

$x = - 3$

$y = 1$

$x = - 3$

$y = 1$

Step 5

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

$(- 3, 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 3, 1)$

Equation Form:

$x = - 3, y = 1$

Step 7