Algebra Examples

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

Step 1

Solve for $x$ in $3 x + 2 y = 4$ .

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Step 1.1

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

$3 x = 4 - 2 y$

$5 x - 4 y = 3$

Step 1.2

Divide each term in $3 x = 4 - 2 y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 4 - 2 y$ by $3$ .

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

$5 x - 4 y = 3$

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.

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

$5 x - 4 y = 3$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$5 x - 4 y = 3$

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

$5 x - 4 y = 3$

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

$5 x - 4 y = 3$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

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

$5 x - 4 y = 3$

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

$5 x - 4 y = 3$

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

$5 x - 4 y = 3$

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

$5 x - 4 y = 3$

Step 2

Replace all occurrences of $x$ with $\frac{4}{3} - \frac{2 y}{3}$ in each equation.

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Step 2.1

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

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 (\frac{4}{3} - \frac{2 y}{3}) - 4 y$ .

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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.

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

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

Step 2.2.1.1.2

Multiply $5 (\frac{4}{3})$ .

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Step 2.2.1.1.2.1

Combine $5$ and $\frac{4}{3}$ .

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

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

Step 2.2.1.1.2.2

Multiply $5$ by $4$ .

$\frac{20}{3} + 5 (- \frac{2 y}{3}) - 4 y = 3$

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

$\frac{20}{3} + 5 (- \frac{2 y}{3}) - 4 y = 3$

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

Step 2.2.1.1.3

Multiply $5 (- \frac{2 y}{3})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $5$ .

$\frac{20}{3} - 5 \frac{2 y}{3} - 4 y = 3$

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

Step 2.2.1.1.3.2

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

$\frac{20}{3} + \frac{- 5 (2 y)}{3} - 4 y = 3$

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

Step 2.2.1.1.3.3

Multiply $2$ by $- 5$ .

$\frac{20}{3} + \frac{- 10 y}{3} - 4 y = 3$

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

$\frac{20}{3} + \frac{- 10 y}{3} - 4 y = 3$

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

Step 2.2.1.1.4

Move the negative in front of the fraction.

$\frac{20}{3} - \frac{10 y}{3} - 4 y = 3$

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

$\frac{20}{3} - \frac{10 y}{3} - 4 y = 3$

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

Step 2.2.1.2

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

$\frac{20}{3} - \frac{10 y}{3} - 4 y \cdot \frac{3}{3} = 3$

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

Step 2.2.1.3

Combine $- 4 y$ and $\frac{3}{3}$ .

$\frac{20}{3} - \frac{10 y}{3} + \frac{- 4 y \cdot 3}{3} = 3$

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

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{20}{3} + \frac{- 10 y - 4 y \cdot 3}{3} = 3$

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

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{20 - 10 y - 4 y \cdot 3}{3} = 3$

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

Step 2.2.1.6

Multiply $3$ by $- 4$ .

$\frac{20 - 10 y - 12 y}{3} = 3$

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

Step 2.2.1.7

Subtract $12 y$ from $- 10 y$ .

$\frac{20 - 22 y}{3} = 3$

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

Step 2.2.1.8

Factor $2$ out of $20 - 22 y$ .

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Step 2.2.1.8.1

Factor $2$ out of $20$ .

$\frac{2 (10) - 22 y}{3} = 3$

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

Step 2.2.1.8.2

Factor $2$ out of $- 22 y$ .

$\frac{2 (10) + 2 (- 11 y)}{3} = 3$

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

Step 2.2.1.8.3

Factor $2$ out of $2 (10) + 2 (- 11 y)$ .

$\frac{2 (10 - 11 y)}{3} = 3$

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

$\frac{2 (10 - 11 y)}{3} = 3$

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

$\frac{2 (10 - 11 y)}{3} = 3$

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

$\frac{2 (10 - 11 y)}{3} = 3$

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

$\frac{2 (10 - 11 y)}{3} = 3$

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

Step 3

Solve for $y$ in $\frac{2 (10 - 11 y)}{3} = 3$ .

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Step 3.1

Multiply both sides by $3$ .

$\frac{2 (10 - 11 y)}{3} \cdot 3 = 3 \cdot 3$

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

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{2 (10 - 11 y)}{3} \cdot 3$ .

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Step 3.2.1.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{2 (10 - 11 y)}{3} \cdot 3 = 3 \cdot 3$

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

Step 3.2.1.1.1.2

Rewrite the expression.

$2 (10 - 11 y) = 3 \cdot 3$

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

$2 (10 - 11 y) = 3 \cdot 3$

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

Step 3.2.1.1.2

Apply the distributive property.

$2 \cdot 10 + 2 (- 11 y) = 3 \cdot 3$

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

Step 3.2.1.1.3

Simplify the expression.

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Step 3.2.1.1.3.1

Multiply $2$ by $10$ .

$20 + 2 (- 11 y) = 3 \cdot 3$

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

Step 3.2.1.1.3.2

Multiply $- 11$ by $2$ .

$20 - 22 y = 3 \cdot 3$

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

Step 3.2.1.1.3.3

Reorder $20$ and $- 22 y$ .

$- 22 y + 20 = 3 \cdot 3$

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

$- 22 y + 20 = 3 \cdot 3$

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

$- 22 y + 20 = 3 \cdot 3$

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

$- 22 y + 20 = 3 \cdot 3$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $3$ by $3$ .

$- 22 y + 20 = 9$

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

$- 22 y + 20 = 9$

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

$- 22 y + 20 = 9$

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $20$ from both sides of the equation.

$- 22 y = 9 - 20$

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

Step 3.3.1.2

Subtract $20$ from $9$ .

$- 22 y = - 11$

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

$- 22 y = - 11$

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

Step 3.3.2

Divide each term in $- 22 y = - 11$ by $- 22$ and simplify.

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Step 3.3.2.1

Divide each term in $- 22 y = - 11$ by $- 22$ .

$\frac{- 22 y}{- 22} = \frac{- 11}{- 22}$

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $- 22$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{- 22 y}{- 22} = \frac{- 11}{- 22}$

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

Step 3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 11}{- 22}$

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

$y = \frac{- 11}{- 22}$

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

$y = \frac{- 11}{- 22}$

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

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Cancel the common factor of $- 11$ and $- 22$ .

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Step 3.3.2.3.1.1

Factor $- 11$ out of $- 11$ .

$y = \frac{- 11 \cdot 1}{- 22}$

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

Step 3.3.2.3.1.2

Cancel the common factors.

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Step 3.3.2.3.1.2.1

Factor $- 11$ out of $- 22$ .

$y = \frac{- 11 \cdot 1}{- 11 \cdot 2}$

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

Step 3.3.2.3.1.2.2

Cancel the common factor.

$y = \frac{- 11 \cdot 1}{- 11 \cdot 2}$

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

Step 3.3.2.3.1.2.3

Rewrite the expression.

$y = \frac{1}{2}$

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

$y = \frac{1}{2}$

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

$y = \frac{1}{2}$

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

$y = \frac{1}{2}$

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

$y = \frac{1}{2}$

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

$y = \frac{1}{2}$

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

$y = \frac{1}{2}$

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

Step 4

Replace all occurrences of $y$ with $\frac{1}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{4}{3} - \frac{2 y}{3}$ with $\frac{1}{2}$ .

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

$y = \frac{1}{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{4}{3} - \frac{2 (\frac{1}{2})}{3}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

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

$y = \frac{1}{2}$

Step 4.2.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.2.1

Factor $2$ out of $- 2$ .

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

$y = \frac{1}{2}$

Step 4.2.1.2.2

Cancel the common factor.

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

$y = \frac{1}{2}$

Step 4.2.1.2.3

Rewrite the expression.

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

$y = \frac{1}{2}$

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

$y = \frac{1}{2}$

Step 4.2.1.3

Simplify the expression.

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Step 4.2.1.3.1

Subtract $1$ from $4$ .

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

$y = \frac{1}{2}$

Step 4.2.1.3.2

Divide $3$ by $3$ .

$x = 1$

$y = \frac{1}{2}$

$x = 1$

$y = \frac{1}{2}$

$x = 1$

$y = \frac{1}{2}$

$x = 1$

$y = \frac{1}{2}$

$x = 1$

$y = \frac{1}{2}$

Step 5

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

$(1, \frac{1}{2})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(1, \frac{1}{2})$

Equation Form:

$x = 1, y = \frac{1}{2}$

Step 7