Linear Algebra Examples

$3 x - 5 y = 51$ , $2 x + 3 y = - 4$

Step 1

Solve for $x$ in $3 x - 5 y = 51$ .

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Step 1.1

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

$3 x = 51 + 5 y$

$2 x + 3 y = - 4$

Step 1.2

Divide each term in $3 x = 51 + 5 y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 51 + 5 y$ by $3$ .

$\frac{3 x}{3} = \frac{51}{3} + \frac{5 y}{3}$

$2 x + 3 y = - 4$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{51}{3} + \frac{5 y}{3}$

$2 x + 3 y = - 4$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{51}{3} + \frac{5 y}{3}$

$2 x + 3 y = - 4$

$x = \frac{51}{3} + \frac{5 y}{3}$

$2 x + 3 y = - 4$

$x = \frac{51}{3} + \frac{5 y}{3}$

$2 x + 3 y = - 4$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $51$ by $3$ .

$x = 17 + \frac{5 y}{3}$

$2 x + 3 y = - 4$

$x = 17 + \frac{5 y}{3}$

$2 x + 3 y = - 4$

$x = 17 + \frac{5 y}{3}$

$2 x + 3 y = - 4$

$x = 17 + \frac{5 y}{3}$

$2 x + 3 y = - 4$

Step 2

Replace all occurrences of $x$ with $17 + \frac{5 y}{3}$ in each equation.

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Step 2.1

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

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

$x = 17 + \frac{5 y}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

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

$x = 17 + \frac{5 y}{3}$

Step 2.2.1.1.2

Multiply $2$ by $17$ .

$34 + 2 (\frac{5 y}{3}) + 3 y = - 4$

$x = 17 + \frac{5 y}{3}$

Step 2.2.1.1.3

Multiply $2 \frac{5 y}{3}$ .

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Step 2.2.1.1.3.1

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

$34 + \frac{2 (5 y)}{3} + 3 y = - 4$

$x = 17 + \frac{5 y}{3}$

Step 2.2.1.1.3.2

Multiply $5$ by $2$ .

$34 + \frac{10 y}{3} + 3 y = - 4$

$x = 17 + \frac{5 y}{3}$

$34 + \frac{10 y}{3} + 3 y = - 4$

$x = 17 + \frac{5 y}{3}$

$34 + \frac{10 y}{3} + 3 y = - 4$

$x = 17 + \frac{5 y}{3}$

Step 2.2.1.2

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

$34 + \frac{10 y}{3} + 3 y \cdot \frac{3}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$34 + \frac{10 y}{3} + \frac{3 y \cdot 3}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$34 + \frac{10 y + 3 y \cdot 3}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

$34 + \frac{10 y + 3 y \cdot 3}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

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 $10 y + 3 y \cdot 3$ .

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Step 2.2.1.4.1.1.1

Factor $y$ out of $10 y$ .

$34 + \frac{y \cdot 10 + 3 y \cdot 3}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

Step 2.2.1.4.1.1.2

Factor $y$ out of $3 y \cdot 3$ .

$34 + \frac{y \cdot 10 + y (3 \cdot 3)}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

Step 2.2.1.4.1.1.3

Factor $y$ out of $y \cdot 10 + y (3 \cdot 3)$ .

$34 + \frac{y (10 + 3 \cdot 3)}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

$34 + \frac{y (10 + 3 \cdot 3)}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

Step 2.2.1.4.1.2

Multiply $3$ by $3$ .

$34 + \frac{y (10 + 9)}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

Step 2.2.1.4.1.3

Add $10$ and $9$ .

$34 + \frac{y \cdot 19}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

$34 + \frac{y \cdot 19}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

Step 2.2.1.4.2

Move $19$ to the left of $y$ .

$34 + \frac{19 y}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

$34 + \frac{19 y}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

$34 + \frac{19 y}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

$34 + \frac{19 y}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

$34 + \frac{19 y}{3} = - 4$

$x = 17 + \frac{5 y}{3}$

Step 3

Solve for $y$ in $34 + \frac{19 y}{3} = - 4$ .

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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 $34$ from both sides of the equation.

$\frac{19 y}{3} = - 4 - 34$

$x = 17 + \frac{5 y}{3}$

Step 3.1.2

Subtract $34$ from $- 4$ .

$\frac{19 y}{3} = - 38$

$x = 17 + \frac{5 y}{3}$

$\frac{19 y}{3} = - 38$

$x = 17 + \frac{5 y}{3}$

Step 3.2

Multiply both sides of the equation by $\frac{3}{19}$ .

$\frac{3}{19} \cdot \frac{19 y}{3} = \frac{3}{19} \cdot - 38$

$x = 17 + \frac{5 y}{3}$

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 $\frac{3}{19} \cdot \frac{19 y}{3}$ .

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Step 3.3.1.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{3}{19} \cdot \frac{19 y}{3} = \frac{3}{19} \cdot - 38$

$x = 17 + \frac{5 y}{3}$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{19} \cdot (19 y) = \frac{3}{19} \cdot - 38$

$x = 17 + \frac{5 y}{3}$

$\frac{1}{19} \cdot (19 y) = \frac{3}{19} \cdot - 38$

$x = 17 + \frac{5 y}{3}$

Step 3.3.1.1.2

Cancel the common factor of $19$ .

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Step 3.3.1.1.2.1

Factor $19$ out of $19 y$ .

$\frac{1}{19} \cdot (19 (y)) = \frac{3}{19} \cdot - 38$

$x = 17 + \frac{5 y}{3}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{19} \cdot (19 y) = \frac{3}{19} \cdot - 38$

$x = 17 + \frac{5 y}{3}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{3}{19} \cdot - 38$

$x = 17 + \frac{5 y}{3}$

$y = \frac{3}{19} \cdot - 38$

$x = 17 + \frac{5 y}{3}$

$y = \frac{3}{19} \cdot - 38$

$x = 17 + \frac{5 y}{3}$

$y = \frac{3}{19} \cdot - 38$

$x = 17 + \frac{5 y}{3}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{3}{19} \cdot - 38$ .

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Step 3.3.2.1.1

Cancel the common factor of $19$ .

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Step 3.3.2.1.1.1

Factor $19$ out of $- 38$ .

$y = \frac{3}{19} \cdot (19 (- 2))$

$x = 17 + \frac{5 y}{3}$

Step 3.3.2.1.1.2

Cancel the common factor.

$y = \frac{3}{19} \cdot (19 \cdot - 2)$

$x = 17 + \frac{5 y}{3}$

Step 3.3.2.1.1.3

Rewrite the expression.

$y = 3 \cdot - 2$

$x = 17 + \frac{5 y}{3}$

$y = 3 \cdot - 2$

$x = 17 + \frac{5 y}{3}$

Step 3.3.2.1.2

Multiply $3$ by $- 2$ .

$y = - 6$

$x = 17 + \frac{5 y}{3}$

$y = - 6$

$x = 17 + \frac{5 y}{3}$

$y = - 6$

$x = 17 + \frac{5 y}{3}$

$y = - 6$

$x = 17 + \frac{5 y}{3}$

$y = - 6$

$x = 17 + \frac{5 y}{3}$

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 $x = 17 + \frac{5 y}{3}$ with $- 6$ .

$x = 17 + \frac{5 (- 6)}{3}$

$y = - 6$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $17 + \frac{5 (- 6)}{3}$ .

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Step 4.2.1.1

Multiply $5$ by $- 6$ .

$x = 17 + \frac{- 30}{3}$

$y = - 6$

Step 4.2.1.2

Divide $- 30$ by $3$ .

$x = 17 - 10$

$y = - 6$

Step 4.2.1.3

Subtract $10$ from $17$ .

$x = 7$

$y = - 6$

$x = 7$

$y = - 6$

$x = 7$

$y = - 6$

$x = 7$

$y = - 6$

Step 5

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

$(7, - 6)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(7, - 6)$

Equation Form:

$x = 7, y = - 6$

Step 7