Linear Algebra Examples

$7 x - 3 y = 2$ , $5 x - 2 y = 3$

Step 1

Solve for $x$ in $7 x - 3 y = 2$ .

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Step 1.1

Add $3 y$ to both sides of the equation.

$7 x = 2 + 3 y$

$5 x - 2 y = 3$

Step 1.2

Divide each term in $7 x = 2 + 3 y$ by $7$ and simplify.

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Step 1.2.1

Divide each term in $7 x = 2 + 3 y$ by $7$ .

$\frac{7 x}{7} = \frac{2}{7} + \frac{3 y}{7}$

$5 x - 2 y = 3$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $7$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{2}{7} + \frac{3 y}{7}$

$5 x - 2 y = 3$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{7} + \frac{3 y}{7}$

$5 x - 2 y = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

$5 x - 2 y = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

$5 x - 2 y = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

$5 x - 2 y = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

$5 x - 2 y = 3$

Step 2

Replace all occurrences of $x$ with $\frac{2}{7} + \frac{3 y}{7}$ in each equation.

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Step 2.1

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

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

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

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

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 2.2.1.1.2

Multiply $5 (\frac{2}{7})$ .

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Step 2.2.1.1.2.1

Combine $5$ and $\frac{2}{7}$ .

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

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 2.2.1.1.2.2

Multiply $5$ by $2$ .

$\frac{10}{7} + 5 (\frac{3 y}{7}) - 2 y = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

$\frac{10}{7} + 5 (\frac{3 y}{7}) - 2 y = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 2.2.1.1.3

Multiply $5 \frac{3 y}{7}$ .

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Step 2.2.1.1.3.1

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

$\frac{10}{7} + \frac{5 (3 y)}{7} - 2 y = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 2.2.1.1.3.2

Multiply $3$ by $5$ .

$\frac{10}{7} + \frac{15 y}{7} - 2 y = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

$\frac{10}{7} + \frac{15 y}{7} - 2 y = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

$\frac{10}{7} + \frac{15 y}{7} - 2 y = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 2.2.1.2

To write $- 2 y$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{10}{7} + \frac{15 y}{7} - 2 y \cdot \frac{7}{7} = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 2.2.1.3

Combine $- 2 y$ and $\frac{7}{7}$ .

$\frac{10}{7} + \frac{15 y}{7} + \frac{- 2 y \cdot 7}{7} = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{10}{7} + \frac{15 y - 2 y \cdot 7}{7} = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{10 + 15 y - 2 y \cdot 7}{7} = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 2.2.1.6

Multiply $7$ by $- 2$ .

$\frac{10 + 15 y - 14 y}{7} = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 2.2.1.7

Subtract $14 y$ from $15 y$ .

$\frac{10 + y}{7} = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

$\frac{10 + y}{7} = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

$\frac{10 + y}{7} = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

$\frac{10 + y}{7} = 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 3

Solve for $y$ in $\frac{10 + y}{7} = 3$ .

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Step 3.1

Multiply both sides of the equation by $7$ .

$7 (\frac{10 + y}{7}) = 7 \cdot 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

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

Cancel the common factor of $7$ .

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Step 3.2.1.1.1

Cancel the common factor.

$7 (\frac{10 + y}{7}) = 7 \cdot 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 3.2.1.1.2

Rewrite the expression.

$10 + y = 7 \cdot 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

$10 + y = 7 \cdot 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

$10 + y = 7 \cdot 3$

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $7$ by $3$ .

$10 + y = 21$

$x = \frac{2}{7} + \frac{3 y}{7}$

$10 + y = 21$

$x = \frac{2}{7} + \frac{3 y}{7}$

$10 + y = 21$

$x = \frac{2}{7} + \frac{3 y}{7}$

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 $10$ from both sides of the equation.

$y = 21 - 10$

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 3.3.2

Subtract $10$ from $21$ .

$y = 11$

$x = \frac{2}{7} + \frac{3 y}{7}$

$y = 11$

$x = \frac{2}{7} + \frac{3 y}{7}$

$y = 11$

$x = \frac{2}{7} + \frac{3 y}{7}$

Step 4

Replace all occurrences of $y$ with $11$ in each equation.

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Step 4.1

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

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

$y = 11$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Combine the numerators over the common denominator.

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

$y = 11$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $3$ by $11$ .

$x = \frac{2 + 33}{7}$

$y = 11$

Step 4.2.1.2.2

Add $2$ and $33$ .

$x = \frac{35}{7}$

$y = 11$

Step 4.2.1.2.3

Divide $35$ by $7$ .

$x = 5$

$y = 11$

$x = 5$

$y = 11$

$x = 5$

$y = 11$

$x = 5$

$y = 11$

$x = 5$

$y = 11$

Step 5

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

$(5, 11)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(5, 11)$

Equation Form:

$x = 5, y = 11$

Step 7