Algebra Examples

$\frac{4}{3} x - \frac{1}{2} y = 2$ $\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1

Solve for $x$ in $\frac{4}{3} x - \frac{1}{2} y = 2$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{4}{3}$ and $x$ .

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.1.2

Combine $y$ and $\frac{1}{2}$ .

$\frac{4 x}{3} - \frac{y}{2} = 2$

$\frac{4}{3} x + \frac{7}{4} y = 11$

$\frac{4 x}{3} - \frac{y}{2} = 2$

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.2

Add $\frac{y}{2}$ to both sides of the equation.

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.3

Multiply both sides of the equation by $\frac{3}{4}$ .

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.4

Simplify both sides of the equation.

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Step 1.4.1

Simplify the left side.

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Step 1.4.1.1

Simplify $\frac{3}{4} \cdot \frac{4 x}{3}$ .

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Step 1.4.1.1.1

Cancel the common factor of $3$ .

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Step 1.4.1.1.1.1

Cancel the common factor.

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.4.1.1.1.2

Rewrite the expression.

$\frac{1}{4} \cdot (4 x) = \frac{3}{4} \cdot (2 + \frac{y}{2})$

$\frac{4}{3} x + \frac{7}{4} y = 11$

$\frac{1}{4} \cdot (4 x) = \frac{3}{4} \cdot (2 + \frac{y}{2})$

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.4.1.1.2

Cancel the common factor of $4$ .

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Step 1.4.1.1.2.1

Factor $4$ out of $4 x$ .

$\frac{1}{4} \cdot (4 (x)) = \frac{3}{4} \cdot (2 + \frac{y}{2})$

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.4.1.1.2.2

Cancel the common factor.

$\frac{1}{4} \cdot (4 x) = \frac{3}{4} \cdot (2 + \frac{y}{2})$

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.4.1.1.2.3

Rewrite the expression.

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify $\frac{3}{4} (2 + \frac{y}{2})$ .

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Step 1.4.2.1.1

Apply the distributive property.

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.4.2.1.2

Cancel the common factor of $2$ .

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Step 1.4.2.1.2.1

Factor $2$ out of $4$ .

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.4.2.1.2.2

Cancel the common factor.

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.4.2.1.2.3

Rewrite the expression.

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.4.2.1.3

Multiply $\frac{3}{4} \cdot \frac{y}{2}$ .

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Step 1.4.2.1.3.1

Multiply $\frac{3}{4}$ by $\frac{y}{2}$ .

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

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.4.2.1.3.2

Multiply $4$ by $2$ .

$x = \frac{3}{2} + \frac{3 y}{8}$

$\frac{4}{3} x + \frac{7}{4} y = 11$

$x = \frac{3}{2} + \frac{3 y}{8}$

$\frac{4}{3} x + \frac{7}{4} y = 11$

$x = \frac{3}{2} + \frac{3 y}{8}$

$\frac{4}{3} x + \frac{7}{4} y = 11$

$x = \frac{3}{2} + \frac{3 y}{8}$

$\frac{4}{3} x + \frac{7}{4} y = 11$

$x = \frac{3}{2} + \frac{3 y}{8}$

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 1.5

Reorder $\frac{3}{2}$ and $\frac{3 y}{8}$ .

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{4}{3} x + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{4}{3} x + \frac{7}{4} y = 11$

Step 2

Replace all occurrences of $x$ with $\frac{3 y}{8} + \frac{3}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $\frac{4}{3} x + \frac{7}{4} y = 11$ with $\frac{3 y}{8} + \frac{3}{2}$ .

$\frac{4}{3} \cdot (\frac{3 y}{8} + \frac{3}{2}) + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{4}{3} \cdot (\frac{3 y}{8} + \frac{3}{2}) + \frac{7}{4} y$ .

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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.

$\frac{4}{3} \cdot \frac{3 y}{8} + \frac{4}{3} \cdot \frac{3}{2} + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.1.2

Cancel the common factor of $4$ .

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Step 2.2.1.1.2.1

Factor $4$ out of $8$ .

$\frac{4}{3} \cdot \frac{3 y}{4 (2)} + \frac{4}{3} \cdot \frac{3}{2} + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.1.2.2

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 y}{4 \cdot 2} + \frac{4}{3} \cdot \frac{3}{2} + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.1.2.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{3 y}{2} + \frac{4}{3} \cdot \frac{3}{2} + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{1}{3} \cdot \frac{3 y}{2} + \frac{4}{3} \cdot \frac{3}{2} + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.1.3

Cancel the common factor of $3$ .

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Step 2.2.1.1.3.1

Factor $3$ out of $3 y$ .

$\frac{1}{3} \cdot \frac{3 (y)}{2} + \frac{4}{3} \cdot \frac{3}{2} + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.1.3.2

Cancel the common factor.

$\frac{1}{3} \cdot \frac{3 y}{2} + \frac{4}{3} \cdot \frac{3}{2} + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.1.3.3

Rewrite the expression.

$\frac{y}{2} + \frac{4}{3} \cdot \frac{3}{2} + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{y}{2} + \frac{4}{3} \cdot \frac{3}{2} + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1

Factor $2$ out of $4$ .

$\frac{y}{2} + \frac{2 (2)}{3} \cdot \frac{3}{2} + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.1.4.2

Cancel the common factor.

$\frac{y}{2} + \frac{2 \cdot 2}{3} \cdot \frac{3}{2} + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.1.4.3

Rewrite the expression.

$\frac{y}{2} + \frac{2}{3} \cdot 3 + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{y}{2} + \frac{2}{3} \cdot 3 + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.1.5

Cancel the common factor of $3$ .

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Step 2.2.1.1.5.1

Cancel the common factor.

$\frac{y}{2} + \frac{2}{3} \cdot 3 + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.1.5.2

Rewrite the expression.

$\frac{y}{2} + 2 + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{y}{2} + 2 + \frac{7}{4} y = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.1.6

Combine $\frac{7}{4}$ and $y$ .

$\frac{y}{2} + 2 + \frac{7 y}{4} = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{y}{2} + 2 + \frac{7 y}{4} = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.2

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

$\frac{y}{2} \cdot \frac{2}{2} + \frac{7 y}{4} + 2 = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.3.1

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

$\frac{y \cdot 2}{2 \cdot 2} + \frac{7 y}{4} + 2 = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.3.2

Multiply $2$ by $2$ .

$\frac{y \cdot 2}{4} + \frac{7 y}{4} + 2 = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{y \cdot 2}{4} + \frac{7 y}{4} + 2 = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{y \cdot 2 + 7 y}{4} + 2 = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.5

Add $y \cdot 2$ and $7 y$ .

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Step 2.2.1.5.1

Reorder $y$ and $2$ .

$\frac{2 \cdot y + 7 y}{4} + 2 = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 2.2.1.5.2

Add $2 \cdot y$ and $7 y$ .

$\frac{9 \cdot y}{4} + 2 = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{9 y}{4} + 2 = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{9 y}{4} + 2 = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{9 y}{4} + 2 = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{9 y}{4} + 2 = 11$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 3

Solve for $y$ in $\frac{9 y}{4} + 2 = 11$ .

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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 $2$ from both sides of the equation.

$\frac{9 y}{4} = 11 - 2$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 3.1.2

Subtract $2$ from $11$ .

$\frac{9 y}{4} = 9$

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{9 y}{4} = 9$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 3.2

Multiply both sides of the equation by $\frac{4}{9}$ .

$\frac{4}{9} \cdot \frac{9 y}{4} = \frac{4}{9} \cdot 9$

$x = \frac{3 y}{8} + \frac{3}{2}$

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 $\frac{4}{9} \cdot \frac{9 y}{4}$ .

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Step 3.3.1.1.1

Cancel the common factor of $4$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{4}{9} \cdot \frac{9 y}{4} = \frac{4}{9} \cdot 9$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{9} \cdot (9 y) = \frac{4}{9} \cdot 9$

$x = \frac{3 y}{8} + \frac{3}{2}$

$\frac{1}{9} \cdot (9 y) = \frac{4}{9} \cdot 9$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 3.3.1.1.2

Cancel the common factor of $9$ .

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Step 3.3.1.1.2.1

Factor $9$ out of $9 y$ .

$\frac{1}{9} \cdot (9 (y)) = \frac{4}{9} \cdot 9$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{9} \cdot (9 y) = \frac{4}{9} \cdot 9$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{4}{9} \cdot 9$

$x = \frac{3 y}{8} + \frac{3}{2}$

$y = \frac{4}{9} \cdot 9$

$x = \frac{3 y}{8} + \frac{3}{2}$

$y = \frac{4}{9} \cdot 9$

$x = \frac{3 y}{8} + \frac{3}{2}$

$y = \frac{4}{9} \cdot 9$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Cancel the common factor of $9$ .

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Step 3.3.2.1.1

Cancel the common factor.

$y = \frac{4}{9} \cdot 9$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 3.3.2.1.2

Rewrite the expression.

$y = 4$

$x = \frac{3 y}{8} + \frac{3}{2}$

$y = 4$

$x = \frac{3 y}{8} + \frac{3}{2}$

$y = 4$

$x = \frac{3 y}{8} + \frac{3}{2}$

$y = 4$

$x = \frac{3 y}{8} + \frac{3}{2}$

$y = 4$

$x = \frac{3 y}{8} + \frac{3}{2}$

Step 4

Replace all occurrences of $y$ with $4$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{3 y}{8} + \frac{3}{2}$ with $4$ .

$x = \frac{3 (4)}{8} + \frac{3}{2}$

$y = 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3 (4)}{8} + \frac{3}{2}$ .

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Step 4.2.1.1

Multiply $3$ by $4$ .

$x = \frac{12}{8} + \frac{3}{2}$

$y = 4$

Step 4.2.1.2

Cancel the common factor of $12$ and $8$ .

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Step 4.2.1.2.1

Factor $4$ out of $12$ .

$x = \frac{4 (3)}{8} + \frac{3}{2}$

$y = 4$

Step 4.2.1.2.2

Cancel the common factors.

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Step 4.2.1.2.2.1

Factor $4$ out of $8$ .

$x = \frac{4 \cdot 3}{4 \cdot 2} + \frac{3}{2}$

$y = 4$

Step 4.2.1.2.2.2

Cancel the common factor.

$x = \frac{4 \cdot 3}{4 \cdot 2} + \frac{3}{2}$

$y = 4$

Step 4.2.1.2.2.3

Rewrite the expression.

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

$y = 4$

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

$y = 4$

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

$y = 4$

Step 4.2.1.3

Combine the numerators over the common denominator.

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

$y = 4$

Step 4.2.1.4

Simplify the expression.

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Step 4.2.1.4.1

Add $3$ and $3$ .

$x = \frac{6}{2}$

$y = 4$

Step 4.2.1.4.2

Divide $6$ by $2$ .

$x = 3$

$y = 4$

$x = 3$

$y = 4$

$x = 3$

$y = 4$

$x = 3$

$y = 4$

$x = 3$

$y = 4$

Step 5

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

$(3, 4)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(3, 4)$

Equation Form:

$x = 3, y = 4$

Step 7