Linear Algebra Examples

$4 x + 3 y = - 10$ , $3 x + 2 y = - 6$

Step 1

Solve for $x$ in $4 x + 3 y = - 10$ .

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Step 1.1

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

$4 x = - 10 - 3 y$

$3 x + 2 y = - 6$

Step 1.2

Divide each term in $4 x = - 10 - 3 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = - 10 - 3 y$ by $4$ .

$\frac{4 x}{4} = \frac{- 10}{4} + \frac{- 3 y}{4}$

$3 x + 2 y = - 6$

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.

$\frac{4 x}{4} = \frac{- 10}{4} + \frac{- 3 y}{4}$

$3 x + 2 y = - 6$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 10}{4} + \frac{- 3 y}{4}$

$3 x + 2 y = - 6$

$x = \frac{- 10}{4} + \frac{- 3 y}{4}$

$3 x + 2 y = - 6$

$x = \frac{- 10}{4} + \frac{- 3 y}{4}$

$3 x + 2 y = - 6$

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 $- 10$ and $4$ .

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Step 1.2.3.1.1.1

Factor $2$ out of $- 10$ .

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

$3 x + 2 y = - 6$

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 $4$ .

$x = \frac{2 \cdot - 5}{2 \cdot 2} + \frac{- 3 y}{4}$

$3 x + 2 y = - 6$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 5}{2 \cdot 2} + \frac{- 3 y}{4}$

$3 x + 2 y = - 6$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{- 5}{2} + \frac{- 3 y}{4}$

$3 x + 2 y = - 6$

$x = \frac{- 5}{2} + \frac{- 3 y}{4}$

$3 x + 2 y = - 6$

$x = \frac{- 5}{2} + \frac{- 3 y}{4}$

$3 x + 2 y = - 6$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{5}{2} + \frac{- 3 y}{4}$

$3 x + 2 y = - 6$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$x = - \frac{5}{2} - \frac{3 y}{4}$

$3 x + 2 y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$3 x + 2 y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$3 x + 2 y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$3 x + 2 y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$3 x + 2 y = - 6$

Step 2

Replace all occurrences of $x$ with $- \frac{5}{2} - \frac{3 y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x + 2 y = - 6$ with $- \frac{5}{2} - \frac{3 y}{4}$ .

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

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

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

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.2

Multiply $3 (- \frac{5}{2})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $3$ .

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

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.2.2

Combine $- 3$ and $\frac{5}{2}$ .

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

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.2.3

Multiply $- 3$ by $5$ .

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

$x = - \frac{5}{2} - \frac{3 y}{4}$

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

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.3

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

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Step 2.2.1.1.3.1

Multiply $- 1$ by $3$ .

$\frac{- 15}{2} - 3 \frac{3 y}{4} + 2 y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.3.2

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

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

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.3.3

Multiply $3$ by $- 3$ .

$\frac{- 15}{2} + \frac{- 9 y}{4} + 2 y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$\frac{- 15}{2} + \frac{- 9 y}{4} + 2 y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

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.

$- \frac{15}{2} + \frac{- 9 y}{4} + 2 y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{15}{2} - \frac{9 y}{4} + 2 y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{15}{2} - \frac{9 y}{4} + 2 y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{15}{2} - \frac{9 y}{4} + 2 y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.2

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

$- \frac{15}{2} - \frac{9 y}{4} + 2 y \cdot \frac{4}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$- \frac{15}{2} - \frac{9 y}{4} + \frac{2 y \cdot 4}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$- \frac{15}{2} + \frac{- 9 y + 2 y \cdot 4}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{15}{2} + \frac{- 9 y + 2 y \cdot 4}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4

Simplify each term.

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Step 2.2.1.4.1

Simplify the numerator.

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Step 2.2.1.4.1.1

Factor $y$ out of $- 9 y + 2 y \cdot 4$ .

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Step 2.2.1.4.1.1.1

Factor $y$ out of $- 9 y$ .

$- \frac{15}{2} + \frac{y \cdot - 9 + 2 y \cdot 4}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4.1.1.2

Factor $y$ out of $2 y \cdot 4$ .

$- \frac{15}{2} + \frac{y \cdot - 9 + y (2 \cdot 4)}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4.1.1.3

Factor $y$ out of $y \cdot - 9 + y (2 \cdot 4)$ .

$- \frac{15}{2} + \frac{y (- 9 + 2 \cdot 4)}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{15}{2} + \frac{y (- 9 + 2 \cdot 4)}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4.1.2

Multiply $2$ by $4$ .

$- \frac{15}{2} + \frac{y (- 9 + 8)}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4.1.3

Add $- 9$ and $8$ .

$- \frac{15}{2} + \frac{y \cdot - 1}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{15}{2} + \frac{y \cdot - 1}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4.2

Move $- 1$ to the left of $y$ .

$- \frac{15}{2} + \frac{- 1 \cdot y}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4.3

Move the negative in front of the fraction.

$- \frac{15}{2} - \frac{y}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{15}{2} - \frac{y}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{15}{2} - \frac{y}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{15}{2} - \frac{y}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{15}{2} - \frac{y}{4} = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3

Solve for $y$ in $- \frac{15}{2} - \frac{y}{4} = - 6$ .

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Step 3.1

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

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Step 3.1.1

Add $\frac{15}{2}$ to both sides of the equation.

$- \frac{y}{4} = - 6 + \frac{15}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.1.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{y}{4} = - 6 \cdot \frac{2}{2} + \frac{15}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.1.3

Combine $- 6$ and $\frac{2}{2}$ .

$- \frac{y}{4} = \frac{- 6 \cdot 2}{2} + \frac{15}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.1.4

Combine the numerators over the common denominator.

$- \frac{y}{4} = \frac{- 6 \cdot 2 + 15}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $- 6$ by $2$ .

$- \frac{y}{4} = \frac{- 12 + 15}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.1.5.2

Add $- 12$ and $15$ .

$- \frac{y}{4} = \frac{3}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{y}{4} = \frac{3}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{y}{4} = \frac{3}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.2

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

$- 4 (- \frac{y}{4}) = - 4 (\frac{3}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

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

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Step 3.3.1.1.1

Cancel the common factor of $4$ .

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Step 3.3.1.1.1.1

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

$- 4 \frac{- y}{4} = - 4 (\frac{3}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.1.2

Factor $4$ out of $- 4$ .

$4 (- 1) (\frac{- y}{4}) = - 4 (\frac{3}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.1.3

Cancel the common factor.

$4 \cdot (- 1 \frac{- y}{4}) = - 4 (\frac{3}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.1.4

Rewrite the expression.

$y = - 4 (\frac{3}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - 4 (\frac{3}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.2

Multiply.

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Step 3.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 y = - 4 (\frac{3}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.2.2

Multiply $y$ by $1$ .

$y = - 4 (\frac{3}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - 4 (\frac{3}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - 4 (\frac{3}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - 4 (\frac{3}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- 4 (\frac{3}{2})$ .

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Step 3.3.2.1.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.1

Factor $2$ out of $- 4$ .

$y = 2 (- 2) (\frac{3}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.2.1.1.2

Cancel the common factor.

$y = 2 \cdot (- 2 (\frac{3}{2}))$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.2.1.1.3

Rewrite the expression.

$y = - 2 \cdot 3$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - 2 \cdot 3$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.2.1.2

Multiply $- 2$ by $3$ .

$y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 4

Replace all occurrences of $y$ with $- 6$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{5}{2} - \frac{3 y}{4}$ with $- 6$ .

$x = - \frac{5}{2} - \frac{3 (- 6)}{4}$

$y = - 6$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{5}{2} - \frac{3 (- 6)}{4}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $- 6$ and $4$ .

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Step 4.2.1.1.1.1

Factor $2$ out of $3 (- 6)$ .

$x = - \frac{5}{2} - \frac{2 (3 \cdot (- 3))}{4}$

$y = - 6$

Step 4.2.1.1.1.2

Cancel the common factors.

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Step 4.2.1.1.1.2.1

Factor $2$ out of $4$ .

$x = - \frac{5}{2} - \frac{2 (3 \cdot (- 3))}{2 (2)}$

$y = - 6$

Step 4.2.1.1.1.2.2

Cancel the common factor.

$x = - \frac{5}{2} - \frac{2 (3 \cdot (- 3))}{2 \cdot 2}$

$y = - 6$

Step 4.2.1.1.1.2.3

Rewrite the expression.

$x = - \frac{5}{2} - \frac{3 \cdot (- 3)}{2}$

$y = - 6$

$x = - \frac{5}{2} - \frac{3 \cdot (- 3)}{2}$

$y = - 6$

$x = - \frac{5}{2} - \frac{3 \cdot (- 3)}{2}$

$y = - 6$

Step 4.2.1.1.2

Multiply $3$ by $- 3$ .

$x = - \frac{5}{2} - \frac{- 9}{2}$

$y = - 6$

Step 4.2.1.1.3

Move the negative in front of the fraction.

$x = - \frac{5}{2} + \frac{9}{2}$

$y = - 6$

Step 4.2.1.1.4

Multiply $- - \frac{9}{2}$ .

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Step 4.2.1.1.4.1

Multiply $- 1$ by $- 1$ .

$x = - \frac{5}{2} + 1 (\frac{9}{2})$

$y = - 6$

Step 4.2.1.1.4.2

Multiply $\frac{9}{2}$ by $1$ .

$x = - \frac{5}{2} + \frac{9}{2}$

$y = - 6$

$x = - \frac{5}{2} + \frac{9}{2}$

$y = - 6$

$x = - \frac{5}{2} + \frac{9}{2}$

$y = - 6$

Step 4.2.1.2

Combine fractions.

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Step 4.2.1.2.1

Combine the numerators over the common denominator.

$x = \frac{- 5 + 9}{2}$

$y = - 6$

Step 4.2.1.2.2

Simplify the expression.

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Step 4.2.1.2.2.1

Add $- 5$ and $9$ .

$x = \frac{4}{2}$

$y = - 6$

Step 4.2.1.2.2.2

Divide $4$ by $2$ .

$x = 2$

$y = - 6$

$x = 2$

$y = - 6$

$x = 2$

$y = - 6$

$x = 2$

$y = - 6$

$x = 2$

$y = - 6$

$x = 2$

$y = - 6$

Step 5

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

$(2, - 6)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, - 6)$

Equation Form:

$x = 2, y = - 6$

Step 7