Linear Algebra Examples

$4 x - 10 y = 6$ , $4 x - 3 y = - 5$

Step 1

Solve for $x$ in $4 x - 10 y = 6$ .

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Step 1.1

Add $10 y$ to both sides of the equation.

$4 x = 6 + 10 y$

$4 x - 3 y = - 5$

Step 1.2

Divide each term in $4 x = 6 + 10 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = 6 + 10 y$ by $4$ .

$\frac{4 x}{4} = \frac{6}{4} + \frac{10 y}{4}$

$4 x - 3 y = - 5$

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.

$\frac{4 x}{4} = \frac{6}{4} + \frac{10 y}{4}$

$4 x - 3 y = - 5$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{4} + \frac{10 y}{4}$

$4 x - 3 y = - 5$

$x = \frac{6}{4} + \frac{10 y}{4}$

$4 x - 3 y = - 5$

$x = \frac{6}{4} + \frac{10 y}{4}$

$4 x - 3 y = - 5$

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 $6$ and $4$ .

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Step 1.2.3.1.1.1

Factor $2$ out of $6$ .

$x = \frac{2 (3)}{4} + \frac{10 y}{4}$

$4 x - 3 y = - 5$

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 $4$ .

$x = \frac{2 \cdot 3}{2 \cdot 2} + \frac{10 y}{4}$

$4 x - 3 y = - 5$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 3}{2 \cdot 2} + \frac{10 y}{4}$

$4 x - 3 y = - 5$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{3}{2} + \frac{10 y}{4}$

$4 x - 3 y = - 5$

$x = \frac{3}{2} + \frac{10 y}{4}$

$4 x - 3 y = - 5$

$x = \frac{3}{2} + \frac{10 y}{4}$

$4 x - 3 y = - 5$

Step 1.2.3.1.2

Cancel the common factor of $10$ and $4$ .

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Step 1.2.3.1.2.1

Factor $2$ out of $10 y$ .

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

$4 x - 3 y = - 5$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $4$ .

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

$4 x - 3 y = - 5$

Step 1.2.3.1.2.2.2

Cancel the common factor.

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

$4 x - 3 y = - 5$

Step 1.2.3.1.2.2.3

Rewrite the expression.

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

$4 x - 3 y = - 5$

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

$4 x - 3 y = - 5$

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

$4 x - 3 y = - 5$

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

$4 x - 3 y = - 5$

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

$4 x - 3 y = - 5$

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

$4 x - 3 y = - 5$

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

$4 x - 3 y = - 5$

Step 2

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

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Step 2.1

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

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

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

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

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 $4$ .

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

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

Step 2.2.1.1.2.2

Cancel the common factor.

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

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

Step 2.2.1.1.2.3

Rewrite the expression.

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

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

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

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

Step 2.2.1.1.3

Multiply $2$ by $3$ .

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

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

Step 2.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1

Factor $2$ out of $4$ .

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

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

Step 2.2.1.1.4.2

Cancel the common factor.

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

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

Step 2.2.1.1.4.3

Rewrite the expression.

$6 + 2 (5 y) - 3 y = - 5$

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

$6 + 2 (5 y) - 3 y = - 5$

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

Step 2.2.1.1.5

Multiply $5$ by $2$ .

$6 + 10 y - 3 y = - 5$

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

$6 + 10 y - 3 y = - 5$

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

Step 2.2.1.2

Subtract $3 y$ from $10 y$ .

$6 + 7 y = - 5$

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

$6 + 7 y = - 5$

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

$6 + 7 y = - 5$

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

$6 + 7 y = - 5$

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

Step 3

Solve for $y$ in $6 + 7 y = - 5$ .

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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

Subtract $6$ from both sides of the equation.

$7 y = - 5 - 6$

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

Step 3.1.2

Subtract $6$ from $- 5$ .

$7 y = - 11$

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

$7 y = - 11$

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

Step 3.2

Divide each term in $7 y = - 11$ by $7$ and simplify.

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Step 3.2.1

Divide each term in $7 y = - 11$ by $7$ .

$\frac{7 y}{7} = \frac{- 11}{7}$

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $7$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{- 11}{7}$

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

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 11}{7}$

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

$y = \frac{- 11}{7}$

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

$y = \frac{- 11}{7}$

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$y = - \frac{11}{7}$

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

$y = - \frac{11}{7}$

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

$y = - \frac{11}{7}$

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

$y = - \frac{11}{7}$

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

Step 4

Replace all occurrences of $y$ with $- \frac{11}{7}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{3}{2} + \frac{5 y}{2}$ with $- \frac{11}{7}$ .

$x = \frac{3}{2} + \frac{5 (- \frac{11}{7})}{2}$

$y = - \frac{11}{7}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3}{2} + \frac{5 (- \frac{11}{7})}{2}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{3 + 5 (- \frac{11}{7})}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Multiply $5 (- \frac{11}{7})$ .

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Step 4.2.1.2.1.1

Multiply $- 1$ by $5$ .

$x = \frac{3 - 5 (\frac{11}{7})}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.2.1.2

Combine $- 5$ and $\frac{11}{7}$ .

$x = \frac{3 + \frac{- 5 \cdot 11}{7}}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.2.1.3

Multiply $- 5$ by $11$ .

$x = \frac{3 + \frac{- 55}{7}}{2}$

$y = - \frac{11}{7}$

$x = \frac{3 + \frac{- 55}{7}}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.2.2

Move the negative in front of the fraction.

$x = \frac{3 - \frac{55}{7}}{2}$

$y = - \frac{11}{7}$

$x = \frac{3 - \frac{55}{7}}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = \frac{3 \cdot \frac{7}{7} - \frac{55}{7}}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.4

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

$x = \frac{\frac{3 \cdot 7}{7} - \frac{55}{7}}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{3 \cdot 7 - 55}{7}}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.6

Simplify the numerator.

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Step 4.2.1.6.1

Multiply $3$ by $7$ .

$x = \frac{\frac{21 - 55}{7}}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.6.2

Subtract $55$ from $21$ .

$x = \frac{\frac{- 34}{7}}{2}$

$y = - \frac{11}{7}$

$x = \frac{\frac{- 34}{7}}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.7

Move the negative in front of the fraction.

$x = \frac{- \frac{34}{7}}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.8

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{34}{7} \cdot \frac{1}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.9

Cancel the common factor of $2$ .

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Step 4.2.1.9.1

Move the leading negative in $- \frac{34}{7}$ into the numerator.

$x = \frac{- 34}{7} \cdot \frac{1}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.9.2

Factor $2$ out of $- 34$ .

$x = \frac{2 (- 17)}{7} \cdot \frac{1}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.9.3

Cancel the common factor.

$x = \frac{2 \cdot - 17}{7} \cdot \frac{1}{2}$

$y = - \frac{11}{7}$

Step 4.2.1.9.4

Rewrite the expression.

$x = \frac{- 17}{7}$

$y = - \frac{11}{7}$

$x = \frac{- 17}{7}$

$y = - \frac{11}{7}$

Step 4.2.1.10

Move the negative in front of the fraction.

$x = - \frac{17}{7}$

$y = - \frac{11}{7}$

$x = - \frac{17}{7}$

$y = - \frac{11}{7}$

$x = - \frac{17}{7}$

$y = - \frac{11}{7}$

$x = - \frac{17}{7}$

$y = - \frac{11}{7}$

Step 5

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

$(- \frac{17}{7}, - \frac{11}{7})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{17}{7}, - \frac{11}{7})$

Equation Form:

$x = - \frac{17}{7}, y = - \frac{11}{7}$

Step 7