Algebra Examples

$\frac{x + 3}{4} + \frac{y - 1}{3} = 1$ , $2 x - y = 12$

Step 1

Simplify $\frac{x + 3}{4} + \frac{y - 1}{3}$ .

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Step 1.1

To write $\frac{x + 3}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x + 3}{4} \cdot \frac{3}{3} + \frac{y - 1}{3} = 1$

$2 x - y = 12$

Step 1.2

To write $\frac{y - 1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x + 3}{4} \cdot \frac{3}{3} + \frac{y - 1}{3} \cdot \frac{4}{4} = 1$

$2 x - y = 12$

Step 1.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{x + 3}{4}$ by $\frac{3}{3}$ .

$\frac{(x + 3) \cdot 3}{4 \cdot 3} + \frac{y - 1}{3} \cdot \frac{4}{4} = 1$

$2 x - y = 12$

Step 1.3.2

Multiply $4$ by $3$ .

$\frac{(x + 3) \cdot 3}{12} + \frac{y - 1}{3} \cdot \frac{4}{4} = 1$

$2 x - y = 12$

Step 1.3.3

Multiply $\frac{y - 1}{3}$ by $\frac{4}{4}$ .

$\frac{(x + 3) \cdot 3}{12} + \frac{(y - 1) \cdot 4}{3 \cdot 4} = 1$

$2 x - y = 12$

Step 1.3.4

Multiply $3$ by $4$ .

$\frac{(x + 3) \cdot 3}{12} + \frac{(y - 1) \cdot 4}{12} = 1$

$2 x - y = 12$

$\frac{(x + 3) \cdot 3}{12} + \frac{(y - 1) \cdot 4}{12} = 1$

$2 x - y = 12$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(x + 3) \cdot 3 + (y - 1) \cdot 4}{12} = 1$

$2 x - y = 12$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{x \cdot 3 + 3 \cdot 3 + (y - 1) \cdot 4}{12} = 1$

$2 x - y = 12$

Step 1.5.2

Move $3$ to the left of $x$ .

$\frac{3 \cdot x + 3 \cdot 3 + (y - 1) \cdot 4}{12} = 1$

$2 x - y = 12$

Step 1.5.3

Multiply $3$ by $3$ .

$\frac{3 \cdot x + 9 + (y - 1) \cdot 4}{12} = 1$

$2 x - y = 12$

Step 1.5.4

Apply the distributive property.

$\frac{3 x + 9 + y \cdot 4 - 1 \cdot 4}{12} = 1$

$2 x - y = 12$

Step 1.5.5

Move $4$ to the left of $y$ .

$\frac{3 x + 9 + 4 \cdot y - 1 \cdot 4}{12} = 1$

$2 x - y = 12$

Step 1.5.6

Multiply $- 1$ by $4$ .

$\frac{3 x + 9 + 4 \cdot y - 4}{12} = 1$

$2 x - y = 12$

Step 1.5.7

Subtract $4$ from $9$ .

$\frac{3 x + 4 y + 5}{12} = 1$

$2 x - y = 12$

$\frac{3 x + 4 y + 5}{12} = 1$

$2 x - y = 12$

$\frac{3 x + 4 y + 5}{12} = 1$

$2 x - y = 12$

Step 2

Multiply both sides by $12$ .

$\frac{3 x + 4 y + 5}{12} \cdot 12 = 1 \cdot 12$

$2 x - y = 12$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $12$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{3 x + 4 y + 5}{12} \cdot 12 = 1 \cdot 12$

$2 x - y = 12$

Step 3.1.1.2

Rewrite the expression.

$3 x + 4 y + 5 = 1 \cdot 12$

$2 x - y = 12$

$3 x + 4 y + 5 = 1 \cdot 12$

$2 x - y = 12$

$3 x + 4 y + 5 = 1 \cdot 12$

$2 x - y = 12$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $12$ by $1$ .

$3 x + 4 y + 5 = 12$

$2 x - y = 12$

$3 x + 4 y + 5 = 12$

$2 x - y = 12$

$3 x + 4 y + 5 = 12$

$2 x - y = 12$

Step 4

Solve for $y$ .

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Step 4.1

Move all terms not containing $y$ to the right side of the equation.

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Step 4.1.1

Subtract $3 x$ from both sides of the equation.

$4 y + 5 = 12 - 3 x$

$2 x - y = 12$

Step 4.1.2

Subtract $5$ from both sides of the equation.

$4 y = 12 - 3 x - 5$

$2 x - y = 12$

Step 4.1.3

Subtract $5$ from $12$ .

$4 y = - 3 x + 7$

$2 x - y = 12$

$4 y = - 3 x + 7$

$2 x - y = 12$

Step 4.2

Divide each term in $4 y = - 3 x + 7$ by $4$ and simplify.

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Step 4.2.1

Divide each term in $4 y = - 3 x + 7$ by $4$ .

$\frac{4 y}{4} = \frac{- 3 x}{4} + \frac{7}{4}$

$2 x - y = 12$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 3 x}{4} + \frac{7}{4}$

$2 x - y = 12$

Step 4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3 x}{4} + \frac{7}{4}$

$2 x - y = 12$

$y = \frac{- 3 x}{4} + \frac{7}{4}$

$2 x - y = 12$

$y = \frac{- 3 x}{4} + \frac{7}{4}$

$2 x - y = 12$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative in front of the fraction.

$y = - \frac{3 x}{4} + \frac{7}{4}$

$2 x - y = 12$

$y = - \frac{3 x}{4} + \frac{7}{4}$

$2 x - y = 12$

$y = - \frac{3 x}{4} + \frac{7}{4}$

$2 x - y = 12$

$y = - \frac{3 x}{4} + \frac{7}{4}$

$2 x - y = 12$

Step 5

Subtract $2 x$ from both sides of the equation.

$- y = 12 - 2 x$

$y = - \frac{3 x}{4} + \frac{7}{4}$

Step 6

Divide each term in $- y = 12 - 2 x$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- y = 12 - 2 x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{12}{- 1} + \frac{- 2 x}{- 1}$

$y = - \frac{3 x}{4} + \frac{7}{4}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{12}{- 1} + \frac{- 2 x}{- 1}$

$y = - \frac{3 x}{4} + \frac{7}{4}$

Step 6.2.2

Divide $y$ by $1$ .

$y = \frac{12}{- 1} + \frac{- 2 x}{- 1}$

$y = - \frac{3 x}{4} + \frac{7}{4}$

$y = \frac{12}{- 1} + \frac{- 2 x}{- 1}$

$y = - \frac{3 x}{4} + \frac{7}{4}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Divide $12$ by $- 1$ .

$y = - 12 + \frac{- 2 x}{- 1}$

$y = - \frac{3 x}{4} + \frac{7}{4}$

Step 6.3.1.2

Move the negative one from the denominator of $\frac{- 2 x}{- 1}$ .

$y = - 12 - 1 \cdot (- 2 x)$

$y = - \frac{3 x}{4} + \frac{7}{4}$

Step 6.3.1.3

Rewrite $- 1 \cdot (- 2 x)$ as $- (- 2 x)$ .

$y = - 12 - (- 2 x)$

$y = - \frac{3 x}{4} + \frac{7}{4}$

Step 6.3.1.4

Multiply $- 2$ by $- 1$ .

$y = - 12 + 2 x$

$y = - \frac{3 x}{4} + \frac{7}{4}$

$y = - 12 + 2 x$

$y = - \frac{3 x}{4} + \frac{7}{4}$

$y = - 12 + 2 x$

$y = - \frac{3 x}{4} + \frac{7}{4}$

$y = - 12 + 2 x$

$y = - \frac{3 x}{4} + \frac{7}{4}$

Step 7

Create a graph to locate the intersection of the equations. The intersection of the system of equations is the solution.

$(5, - 2)$

Step 8