Algebra Examples

$- 3 x + y = 6$ $a x + 2 y = 4$

Step 1

Solve the equation for $x$ .

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Step 1.1

Subtract $y$ from both sides of the equation.

$- 3 x = 6 - y$

$a x + 2 y = 4$

Step 1.2

Divide each term in $- 3 x = 6 - y$ by $- 3$ and simplify.

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Step 1.2.1

Divide each term in $- 3 x = 6 - y$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{6}{- 3} + \frac{- y}{- 3}$

$a x + 2 y = 4$

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.

$\frac{- 3 x}{- 3} = \frac{6}{- 3} + \frac{- y}{- 3}$

$a x + 2 y = 4$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{- 3} + \frac{- y}{- 3}$

$a x + 2 y = 4$

$x = \frac{6}{- 3} + \frac{- y}{- 3}$

$a x + 2 y = 4$

$x = \frac{6}{- 3} + \frac{- y}{- 3}$

$a x + 2 y = 4$

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 $6$ by $- 3$ .

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

$a x + 2 y = 4$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

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

$a x + 2 y = 4$

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

$a x + 2 y = 4$

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

$a x + 2 y = 4$

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

$a x + 2 y = 4$

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

$a x + 2 y = 4$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

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

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

Step 2.1.2

Move $- 2$ to the left of $a$ .

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

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

Step 2.1.3

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

$- 2 a + \frac{a y}{3} + 2 y = 4$

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

$- 2 a + \frac{a y}{3} + 2 y = 4$

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

Step 2.2

Add $2 a$ to both sides of the equation.

$\frac{a y}{3} + 2 y = 4 + 2 a$

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

Step 2.3

Factor $y$ out of $\frac{a y}{3} + 2 y$ .

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Step 2.3.1

Factor $y$ out of $\frac{a y}{3}$ .

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

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

Step 2.3.2

Factor $y$ out of $2 y$ .

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

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

Step 2.3.3

Factor $y$ out of $y \frac{a}{3} + y \cdot 2$ .

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

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

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

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

Step 2.4

Divide each term in $y (\frac{a}{3} + 2) = 4 + 2 a$ by $\frac{a}{3} + 2$ and simplify.

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Step 2.4.1

Divide each term in $y (\frac{a}{3} + 2) = 4 + 2 a$ by $\frac{a}{3} + 2$ .

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

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $\frac{a}{3} + 2$ .

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Step 2.4.2.1.1

Cancel the common factor.

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

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

Step 2.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{4}{\frac{a}{3} + 2} + \frac{2 a}{\frac{a}{3} + 2}$

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

$y = \frac{4}{\frac{a}{3} + 2} + \frac{2 a}{\frac{a}{3} + 2}$

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

$y = \frac{4}{\frac{a}{3} + 2} + \frac{2 a}{\frac{a}{3} + 2}$

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Combine the numerators over the common denominator.

$y = \frac{4 + 2 a}{\frac{a}{3} + 2}$

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

Step 2.4.3.2

Multiply the numerator and denominator of the fraction by $3$ .

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Step 2.4.3.2.1

Multiply $\frac{4 + 2 a}{\frac{a}{3} + 2}$ by $\frac{3}{3}$ .

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

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

Step 2.4.3.2.2

Combine.

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

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

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

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

Step 2.4.3.3

Apply the distributive property.

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

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

Step 2.4.3.4

Cancel the common factor of $3$ .

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Step 2.4.3.4.1

Cancel the common factor.

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

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

Step 2.4.3.4.2

Rewrite the expression.

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

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

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

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

Step 2.4.3.5

Simplify the numerator.

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Step 2.4.3.5.1

Factor $3$ out of $3 \cdot 4 + 3 \cdot 2 a$ .

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Step 2.4.3.5.1.1

Factor $3$ out of $3 \cdot 4$ .

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

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

Step 2.4.3.5.1.2

Factor $3$ out of $3 \cdot 2 a$ .

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

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

Step 2.4.3.5.1.3

Factor $3$ out of $3 (4) + 3 (2 a)$ .

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

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

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

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

Step 2.4.3.5.2

Factor $2$ out of $4 + 2 a$ .

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Step 2.4.3.5.2.1

Factor $2$ out of $4$ .

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

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

Step 2.4.3.5.2.2

Factor $2$ out of $2 \cdot 2 + 2 a$ .

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

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

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

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

Step 2.4.3.5.3

Multiply $3$ by $2$ .

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

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

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

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

Step 2.4.3.6

Multiply $3$ by $2$ .

$y = \frac{6 (2 + a)}{a + 6}$

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

$y = \frac{6 (2 + a)}{a + 6}$

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

$y = \frac{6 (2 + a)}{a + 6}$

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

$y = \frac{6 (2 + a)}{a + 6}$

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

Step 3

Simplify the right side.

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Step 3.1

Simplify $- 2 + \frac{\frac{6 (2 + a)}{a + 6}}{3}$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = - 2 + \frac{6 (2 + a)}{a + 6} \cdot \frac{1}{3}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.1.2

Cancel the common factor of $3$ .

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Step 3.1.1.2.1

Factor $3$ out of $6 (2 + a)$ .

$x = - 2 + \frac{3 (2 (2 + a))}{a + 6} \cdot \frac{1}{3}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.1.2.2

Cancel the common factor.

$x = - 2 + \frac{3 (2 (2 + a))}{a + 6} \cdot \frac{1}{3}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.1.2.3

Rewrite the expression.

$x = - 2 + \frac{2 (2 + a)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

$x = - 2 + \frac{2 (2 + a)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

$x = - 2 + \frac{2 (2 + a)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{a + 6}{a + 6}$ .

$x = - 2 \cdot \frac{a + 6}{a + 6} + \frac{2 (2 + a)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.3

Simplify terms.

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Step 3.1.3.1

Combine $- 2$ and $\frac{a + 6}{a + 6}$ .

$x = \frac{- 2 (a + 6)}{a + 6} + \frac{2 (2 + a)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.3.2

Combine the numerators over the common denominator.

$x = \frac{- 2 (a + 6) + 2 (2 + a)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

$x = \frac{- 2 (a + 6) + 2 (2 + a)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.4

Simplify the numerator.

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Step 3.1.4.1

Factor $2$ out of $- 2 (a + 6) + 2 (2 + a)$ .

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Step 3.1.4.1.1

Factor $2$ out of $- 2 (a + 6)$ .

$x = \frac{2 (- (a + 6)) + 2 (2 + a)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.4.1.2

Factor $2$ out of $2 (- (a + 6)) + 2 (2 + a)$ .

$x = \frac{2 (- (a + 6) + 2 + a)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

$x = \frac{2 (- (a + 6) + 2 + a)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.4.2

Apply the distributive property.

$x = \frac{2 (- a - 1 \cdot 6 + 2 + a)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.4.3

Multiply $- 1$ by $6$ .

$x = \frac{2 (- a - 6 + 2 + a)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.4.4

Add $- a$ and $a$ .

$x = \frac{2 (0 - 6 + 2)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.4.5

Subtract $6$ from $0$ .

$x = \frac{2 (- 6 + 2)}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.4.6

Add $- 6$ and $2$ .

$x = \frac{2 \cdot - 4}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

$x = \frac{2 \cdot - 4}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.5

Simplify the expression.

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Step 3.1.5.1

Multiply $2$ by $- 4$ .

$x = \frac{- 8}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

Step 3.1.5.2

Move the negative in front of the fraction.

$x = - \frac{8}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

$x = - \frac{8}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

$x = - \frac{8}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$

$x = - \frac{8}{a + 6}$

$y = \frac{6 (2 + a)}{a + 6}$