Linear Algebra Examples

$4 x + 3 y + 7 = 0$ , $5 x - 2 y + 3 = 0$

Step 1

Solve for $x$ in $4 x + 3 y + 7 = 0$ .

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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 $3 y$ from both sides of the equation.

$4 x + 7 = - 3 y$

$5 x - 2 y + 3 = 0$

Step 1.1.2

Subtract $7$ from both sides of the equation.

$4 x = - 3 y - 7$

$5 x - 2 y + 3 = 0$

$4 x = - 3 y - 7$

$5 x - 2 y + 3 = 0$

Step 1.2

Divide each term in $4 x = - 3 y - 7$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = - 3 y - 7$ by $4$ .

$\frac{4 x}{4} = \frac{- 3 y}{4} + \frac{- 7}{4}$

$5 x - 2 y + 3 = 0$

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.

$\frac{4 x}{4} = \frac{- 3 y}{4} + \frac{- 7}{4}$

$5 x - 2 y + 3 = 0$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3 y}{4} + \frac{- 7}{4}$

$5 x - 2 y + 3 = 0$

$x = \frac{- 3 y}{4} + \frac{- 7}{4}$

$5 x - 2 y + 3 = 0$

$x = \frac{- 3 y}{4} + \frac{- 7}{4}$

$5 x - 2 y + 3 = 0$

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

Move the negative in front of the fraction.

$x = - \frac{3 y}{4} + \frac{- 7}{4}$

$5 x - 2 y + 3 = 0$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{3 y}{4} - \frac{7}{4}$

$5 x - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$5 x - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$5 x - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$5 x - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$5 x - 2 y + 3 = 0$

Step 2

Replace all occurrences of $x$ with $- \frac{3 y}{4} - \frac{7}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $5 x - 2 y + 3 = 0$ with $- \frac{3 y}{4} - \frac{7}{4}$ .

$5 (- \frac{3 y}{4} - \frac{7}{4}) - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 (- \frac{3 y}{4} - \frac{7}{4}) - 2 y + 3$ .

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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.

$5 (- \frac{3 y}{4}) + 5 (- \frac{7}{4}) - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.1.2

Multiply $5 (- \frac{3 y}{4})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $5$ .

$- 5 \frac{3 y}{4} + 5 (- \frac{7}{4}) - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.1.2.2

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

$\frac{- 5 (3 y)}{4} + 5 (- \frac{7}{4}) - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.1.2.3

Multiply $3$ by $- 5$ .

$\frac{- 15 y}{4} + 5 (- \frac{7}{4}) - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$\frac{- 15 y}{4} + 5 (- \frac{7}{4}) - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.1.3

Multiply $5 (- \frac{7}{4})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $5$ .

$\frac{- 15 y}{4} - 5 (\frac{7}{4}) - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.1.3.2

Combine $- 5$ and $\frac{7}{4}$ .

$\frac{- 15 y}{4} + \frac{- 5 \cdot 7}{4} - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.1.3.3

Multiply $- 5$ by $7$ .

$\frac{- 15 y}{4} + \frac{- 35}{4} - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$\frac{- 15 y}{4} + \frac{- 35}{4} - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

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.

$- \frac{15 y}{4} + \frac{- 35}{4} - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{15 y}{4} - \frac{35}{4} - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$- \frac{15 y}{4} - \frac{35}{4} - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$- \frac{15 y}{4} - \frac{35}{4} - 2 y + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.2

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

$- \frac{15 y}{4} - 2 y \cdot \frac{4}{4} - \frac{35}{4} + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.3

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

$- \frac{15 y}{4} + \frac{- 2 y \cdot 4}{4} - \frac{35}{4} + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{- 15 y - 2 y \cdot 4}{4} - \frac{35}{4} + 3 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$3 + \frac{- 15 y - 2 y \cdot 4 - 35}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.6

Multiply $4$ by $- 2$ .

$3 + \frac{- 15 y - 8 y - 35}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.7

Subtract $8 y$ from $- 15 y$ .

$3 + \frac{- 23 y - 35}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.8

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$3 \cdot \frac{4}{4} + \frac{- 23 y - 35}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.9

Simplify terms.

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Step 2.2.1.9.1

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

$\frac{3 \cdot 4}{4} + \frac{- 23 y - 35}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.9.2

Combine the numerators over the common denominator.

$\frac{3 \cdot 4 - 23 y - 35}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$\frac{3 \cdot 4 - 23 y - 35}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.10

Simplify the numerator.

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Step 2.2.1.10.1

Multiply $3$ by $4$ .

$\frac{12 - 23 y - 35}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.10.2

Subtract $35$ from $12$ .

$\frac{- 23 y - 23}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.10.3

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

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Step 2.2.1.10.3.1

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

$\frac{23 (- y) - 23}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.10.3.2

Factor $23$ out of $- 23$ .

$\frac{23 (- y) + 23 (- 1)}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.10.3.3

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

$\frac{23 (- y - 1)}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$\frac{23 (- y - 1)}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$\frac{23 (- y - 1)}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.11

Simplify with factoring out.

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Step 2.2.1.11.1

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

$\frac{23 (- (y) - 1)}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.11.2

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

$\frac{23 (- (y) - 1 \cdot 1)}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.11.3

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

$\frac{23 (- (y + 1))}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.11.4

Simplify the expression.

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Step 2.2.1.11.4.1

Rewrite $- (y + 1)$ as $- 1 (y + 1)$ .

$\frac{23 (- 1 (y + 1))}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 2.2.1.11.4.2

Move the negative in front of the fraction.

$- \frac{23 (y + 1)}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$- \frac{23 (y + 1)}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$- \frac{23 (y + 1)}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$- \frac{23 (y + 1)}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$- \frac{23 (y + 1)}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$- \frac{23 (y + 1)}{4} = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 3

Solve for $y$ in $- \frac{23 (y + 1)}{4} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$23 (y + 1) = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 3.2

Solve the equation for $y$ .

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Step 3.2.1

Divide each term in $23 (y + 1) = 0$ by $23$ and simplify.

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Step 3.2.1.1

Divide each term in $23 (y + 1) = 0$ by $23$ .

$\frac{23 (y + 1)}{23} = \frac{0}{23}$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 3.2.1.2

Simplify the left side.

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Step 3.2.1.2.1

Cancel the common factor of $23$ .

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Step 3.2.1.2.1.1

Cancel the common factor.

$\frac{23 (y + 1)}{23} = \frac{0}{23}$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 3.2.1.2.1.2

Divide $y + 1$ by $1$ .

$y + 1 = \frac{0}{23}$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$y + 1 = \frac{0}{23}$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$y + 1 = \frac{0}{23}$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 3.2.1.3

Simplify the right side.

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Step 3.2.1.3.1

Divide $0$ by $23$ .

$y + 1 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$y + 1 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$y + 1 = 0$

$x = - \frac{3 y}{4} - \frac{7}{4}$

Step 3.2.2

Subtract $1$ from both sides of the equation.

$y = - 1$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$y = - 1$

$x = - \frac{3 y}{4} - \frac{7}{4}$

$y = - 1$

$x = - \frac{3 y}{4} - \frac{7}{4}$

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 $x = - \frac{3 y}{4} - \frac{7}{4}$ with $- 1$ .

$x = - \frac{3 (- 1)}{4} - \frac{7}{4}$

$y = - 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{3 (- 1)}{4} - \frac{7}{4}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 3 \cdot - 1 - 7}{4}$

$y = - 1$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $- 3$ by $- 1$ .

$x = \frac{3 - 7}{4}$

$y = - 1$

Step 4.2.1.2.2

Subtract $7$ from $3$ .

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

$y = - 1$

Step 4.2.1.2.3

Divide $- 4$ by $4$ .

$x = - 1$

$y = - 1$

$x = - 1$

$y = - 1$

$x = - 1$

$y = - 1$

$x = - 1$

$y = - 1$

$x = - 1$

$y = - 1$

Step 5

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

$(- 1, - 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 1, - 1)$

Equation Form:

$x = - 1, y = - 1$

Step 7