Algebra Examples

$- 2 x - 3 y = 48$ and $- x + 2 y = - 11$

Step 1

Solve for $x$ in $- 2 x - 3 y = 48$ .

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Step 1.1

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

$- 2 x = 48 + 3 y$

$- x + 2 y = - 11$

Step 1.2

Divide each term in $- 2 x = 48 + 3 y$ by $- 2$ and simplify.

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Step 1.2.1

Divide each term in $- 2 x = 48 + 3 y$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{48}{- 2} + \frac{3 y}{- 2}$

$- x + 2 y = - 11$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{48}{- 2} + \frac{3 y}{- 2}$

$- x + 2 y = - 11$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{48}{- 2} + \frac{3 y}{- 2}$

$- x + 2 y = - 11$

$x = \frac{48}{- 2} + \frac{3 y}{- 2}$

$- x + 2 y = - 11$

$x = \frac{48}{- 2} + \frac{3 y}{- 2}$

$- x + 2 y = - 11$

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

Divide $48$ by $- 2$ .

$x = - 24 + \frac{3 y}{- 2}$

$- x + 2 y = - 11$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - 24 - \frac{3 y}{2}$

$- x + 2 y = - 11$

$x = - 24 - \frac{3 y}{2}$

$- x + 2 y = - 11$

$x = - 24 - \frac{3 y}{2}$

$- x + 2 y = - 11$

$x = - 24 - \frac{3 y}{2}$

$- x + 2 y = - 11$

$x = - 24 - \frac{3 y}{2}$

$- x + 2 y = - 11$

Step 2

Replace all occurrences of $x$ with $- 24 - \frac{3 y}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- x + 2 y = - 11$ with $- 24 - \frac{3 y}{2}$ .

$- (- 24 - \frac{3 y}{2}) + 2 y = - 11$

$x = - 24 - \frac{3 y}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- (- 24 - \frac{3 y}{2}) + 2 y$ .

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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.

$24 + \frac{3 y}{2} + 2 y = - 11$

$x = - 24 - \frac{3 y}{2}$

Step 2.2.1.1.2

Multiply $- 1$ by $- 24$ .

$24 + \frac{3 y}{2} + 2 y = - 11$

$x = - 24 - \frac{3 y}{2}$

Step 2.2.1.1.3

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

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Step 2.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$24 + 1 (\frac{3 y}{2}) + 2 y = - 11$

$x = - 24 - \frac{3 y}{2}$

Step 2.2.1.1.3.2

Multiply $\frac{3 y}{2}$ by $1$ .

$24 + \frac{3 y}{2} + 2 y = - 11$

$x = - 24 - \frac{3 y}{2}$

$24 + \frac{3 y}{2} + 2 y = - 11$

$x = - 24 - \frac{3 y}{2}$

$24 + \frac{3 y}{2} + 2 y = - 11$

$x = - 24 - \frac{3 y}{2}$

Step 2.2.1.2

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

$24 + \frac{3 y}{2} + 2 y \cdot \frac{2}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$24 + \frac{3 y}{2} + \frac{2 y \cdot 2}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$24 + \frac{3 y + 2 y \cdot 2}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

$24 + \frac{3 y + 2 y \cdot 2}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

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 $3 y + 2 y \cdot 2$ .

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Step 2.2.1.4.1.1.1

Factor $y$ out of $3 y$ .

$24 + \frac{y \cdot 3 + 2 y \cdot 2}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

Step 2.2.1.4.1.1.2

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

$24 + \frac{y \cdot 3 + y (2 \cdot 2)}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

Step 2.2.1.4.1.1.3

Factor $y$ out of $y \cdot 3 + y (2 \cdot 2)$ .

$24 + \frac{y (3 + 2 \cdot 2)}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

$24 + \frac{y (3 + 2 \cdot 2)}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

Step 2.2.1.4.1.2

Multiply $2$ by $2$ .

$24 + \frac{y (3 + 4)}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

Step 2.2.1.4.1.3

Add $3$ and $4$ .

$24 + \frac{y \cdot 7}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

$24 + \frac{y \cdot 7}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

Step 2.2.1.4.2

Move $7$ to the left of $y$ .

$24 + \frac{7 y}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

$24 + \frac{7 y}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

$24 + \frac{7 y}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

$24 + \frac{7 y}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

$24 + \frac{7 y}{2} = - 11$

$x = - 24 - \frac{3 y}{2}$

Step 3

Solve for $y$ in $24 + \frac{7 y}{2} = - 11$ .

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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 $24$ from both sides of the equation.

$\frac{7 y}{2} = - 11 - 24$

$x = - 24 - \frac{3 y}{2}$

Step 3.1.2

Subtract $24$ from $- 11$ .

$\frac{7 y}{2} = - 35$

$x = - 24 - \frac{3 y}{2}$

$\frac{7 y}{2} = - 35$

$x = - 24 - \frac{3 y}{2}$

Step 3.2

Multiply both sides of the equation by $\frac{2}{7}$ .

$\frac{2}{7} \cdot \frac{7 y}{2} = \frac{2}{7} \cdot - 35$

$x = - 24 - \frac{3 y}{2}$

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 $\frac{2}{7} \cdot \frac{7 y}{2}$ .

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Step 3.3.1.1.1

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{2}{7} \cdot \frac{7 y}{2} = \frac{2}{7} \cdot - 35$

$x = - 24 - \frac{3 y}{2}$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{7} \cdot (7 y) = \frac{2}{7} \cdot - 35$

$x = - 24 - \frac{3 y}{2}$

$\frac{1}{7} \cdot (7 y) = \frac{2}{7} \cdot - 35$

$x = - 24 - \frac{3 y}{2}$

Step 3.3.1.1.2

Cancel the common factor of $7$ .

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Step 3.3.1.1.2.1

Factor $7$ out of $7 y$ .

$\frac{1}{7} \cdot (7 (y)) = \frac{2}{7} \cdot - 35$

$x = - 24 - \frac{3 y}{2}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{7} \cdot (7 y) = \frac{2}{7} \cdot - 35$

$x = - 24 - \frac{3 y}{2}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{2}{7} \cdot - 35$

$x = - 24 - \frac{3 y}{2}$

$y = \frac{2}{7} \cdot - 35$

$x = - 24 - \frac{3 y}{2}$

$y = \frac{2}{7} \cdot - 35$

$x = - 24 - \frac{3 y}{2}$

$y = \frac{2}{7} \cdot - 35$

$x = - 24 - \frac{3 y}{2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{2}{7} \cdot - 35$ .

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Step 3.3.2.1.1

Cancel the common factor of $7$ .

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Step 3.3.2.1.1.1

Factor $7$ out of $- 35$ .

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

$x = - 24 - \frac{3 y}{2}$

Step 3.3.2.1.1.2

Cancel the common factor.

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

$x = - 24 - \frac{3 y}{2}$

Step 3.3.2.1.1.3

Rewrite the expression.

$y = 2 \cdot - 5$

$x = - 24 - \frac{3 y}{2}$

$y = 2 \cdot - 5$

$x = - 24 - \frac{3 y}{2}$

Step 3.3.2.1.2

Multiply $2$ by $- 5$ .

$y = - 10$

$x = - 24 - \frac{3 y}{2}$

$y = - 10$

$x = - 24 - \frac{3 y}{2}$

$y = - 10$

$x = - 24 - \frac{3 y}{2}$

$y = - 10$

$x = - 24 - \frac{3 y}{2}$

$y = - 10$

$x = - 24 - \frac{3 y}{2}$

Step 4

Replace all occurrences of $y$ with $- 10$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - 24 - \frac{3 y}{2}$ with $- 10$ .

$x = - 24 - \frac{3 (- 10)}{2}$

$y = - 10$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 24 - \frac{3 (- 10)}{2}$ .

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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 $- 10$ and $2$ .

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Step 4.2.1.1.1.1

Factor $2$ out of $3 (- 10)$ .

$x = - 24 - \frac{2 (3 \cdot (- 5))}{2}$

$y = - 10$

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 $2$ .

$x = - 24 - \frac{2 (3 \cdot (- 5))}{2 (1)}$

$y = - 10$

Step 4.2.1.1.1.2.2

Cancel the common factor.

$x = - 24 - \frac{2 (3 \cdot (- 5))}{2 \cdot 1}$

$y = - 10$

Step 4.2.1.1.1.2.3

Rewrite the expression.

$x = - 24 - \frac{3 \cdot (- 5)}{1}$

$y = - 10$

Step 4.2.1.1.1.2.4

Divide $3 \cdot (- 5)$ by $1$ .

$x = - 24 - (3 \cdot (- 5))$

$y = - 10$

$x = - 24 - (3 \cdot (- 5))$

$y = - 10$

$x = - 24 - (3 \cdot (- 5))$

$y = - 10$

Step 4.2.1.1.2

Multiply $- (3 \cdot (- 5))$ .

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Step 4.2.1.1.2.1

Multiply $3$ by $- 5$ .

$x = - 24 + 15$

$y = - 10$

Step 4.2.1.1.2.2

Multiply $- 1$ by $- 15$ .

$x = - 24 + 15$

$y = - 10$

$x = - 24 + 15$

$y = - 10$

$x = - 24 + 15$

$y = - 10$

Step 4.2.1.2

Add $- 24$ and $15$ .

$x = - 9$

$y = - 10$

$x = - 9$

$y = - 10$

$x = - 9$

$y = - 10$

$x = - 9$

$y = - 10$

Step 5

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

$(- 9, - 10)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 9, - 10)$

Equation Form:

$x = - 9, y = - 10$

Step 7