Algebra Examples

$5 x - 3 y = - \frac{7}{2}$ $8 x + 3 y = 10$

Step 1

Solve for $x$ in $5 x - 3 y = - \frac{7}{2}$ .

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Step 1.1

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

$5 x = - \frac{7}{2} + 3 y$

$8 x + 3 y = 10$

Step 1.2

Divide each term in $5 x = - \frac{7}{2} + 3 y$ by $5$ and simplify.

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Step 1.2.1

Divide each term in $5 x = - \frac{7}{2} + 3 y$ by $5$ .

$\frac{5 x}{5} = \frac{- \frac{7}{2}}{5} + \frac{3 y}{5}$

$8 x + 3 y = 10$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- \frac{7}{2}}{5} + \frac{3 y}{5}$

$8 x + 3 y = 10$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{7}{2}}{5} + \frac{3 y}{5}$

$8 x + 3 y = 10$

$x = \frac{- \frac{7}{2}}{5} + \frac{3 y}{5}$

$8 x + 3 y = 10$

$x = \frac{- \frac{7}{2}}{5} + \frac{3 y}{5}$

$8 x + 3 y = 10$

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

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{7}{2} \cdot \frac{1}{5} + \frac{3 y}{5}$

$8 x + 3 y = 10$

Step 1.2.3.1.2

Multiply $- \frac{7}{2} \cdot \frac{1}{5}$ .

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Step 1.2.3.1.2.1

Multiply $\frac{1}{5}$ by $\frac{7}{2}$ .

$x = - \frac{7}{5 \cdot 2} + \frac{3 y}{5}$

$8 x + 3 y = 10$

Step 1.2.3.1.2.2

Multiply $5$ by $2$ .

$x = - \frac{7}{10} + \frac{3 y}{5}$

$8 x + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$8 x + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$8 x + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$8 x + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$8 x + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$8 x + 3 y = 10$

Step 2

Replace all occurrences of $x$ with $- \frac{7}{10} + \frac{3 y}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $8 x + 3 y = 10$ with $- \frac{7}{10} + \frac{3 y}{5}$ .

$8 (- \frac{7}{10} + \frac{3 y}{5}) + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $8 (- \frac{7}{10} + \frac{3 y}{5}) + 3 y$ .

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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.

$8 (- \frac{7}{10}) + 8 (\frac{3 y}{5}) + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{7}{10}$ into the numerator.

$8 (\frac{- 7}{10}) + 8 (\frac{3 y}{5}) + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.1.2.2

Factor $2$ out of $8$ .

$2 (4) (\frac{- 7}{10}) + 8 (\frac{3 y}{5}) + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.1.2.3

Factor $2$ out of $10$ .

$2 \cdot (4 (\frac{- 7}{2 \cdot 5})) + 8 (\frac{3 y}{5}) + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.1.2.4

Cancel the common factor.

$2 \cdot (4 (\frac{- 7}{2 \cdot 5})) + 8 (\frac{3 y}{5}) + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.1.2.5

Rewrite the expression.

$4 (\frac{- 7}{5}) + 8 (\frac{3 y}{5}) + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$4 (\frac{- 7}{5}) + 8 (\frac{3 y}{5}) + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.1.3

Combine $4$ and $\frac{- 7}{5}$ .

$\frac{4 \cdot - 7}{5} + 8 (\frac{3 y}{5}) + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.1.4

Multiply $4$ by $- 7$ .

$\frac{- 28}{5} + 8 (\frac{3 y}{5}) + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.1.5

Multiply $8 \frac{3 y}{5}$ .

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Step 2.2.1.1.5.1

Combine $8$ and $\frac{3 y}{5}$ .

$\frac{- 28}{5} + \frac{8 (3 y)}{5} + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.1.5.2

Multiply $3$ by $8$ .

$\frac{- 28}{5} + \frac{24 y}{5} + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$\frac{- 28}{5} + \frac{24 y}{5} + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.1.6

Move the negative in front of the fraction.

$- \frac{28}{5} + \frac{24 y}{5} + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$- \frac{28}{5} + \frac{24 y}{5} + 3 y = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.2

To write $3 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{28}{5} + \frac{24 y}{5} + 3 y \cdot \frac{5}{5} = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.3

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

$- \frac{28}{5} + \frac{24 y}{5} + \frac{3 y \cdot 5}{5} = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{28}{5} + \frac{24 y + 3 y \cdot 5}{5} = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 28 + 24 y + 3 y \cdot 5}{5} = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.6

Multiply $5$ by $3$ .

$\frac{- 28 + 24 y + 15 y}{5} = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.7

Add $24 y$ and $15 y$ .

$\frac{- 28 + 39 y}{5} = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.8

Rewrite $- 28$ as $- 1 (28)$ .

$\frac{- 1 \cdot 28 + 39 y}{5} = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.9

Factor $- 1$ out of $39 y$ .

$\frac{- 1 \cdot 28 - (- 39 y)}{5} = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.10

Factor $- 1$ out of $- 1 (28) - (- 39 y)$ .

$\frac{- 1 (28 - 39 y)}{5} = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 2.2.1.11

Move the negative in front of the fraction.

$- \frac{28 - 39 y}{5} = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$- \frac{28 - 39 y}{5} = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$- \frac{28 - 39 y}{5} = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$- \frac{28 - 39 y}{5} = 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3

Solve for $y$ in $- \frac{28 - 39 y}{5} = 10$ .

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Step 3.1

Multiply both sides of the equation by $- 5$ .

$- 5 (- \frac{28 - 39 y}{5}) = - 5 \cdot 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $- 5 (- \frac{28 - 39 y}{5})$ .

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Step 3.2.1.1.1

Cancel the common factor of $5$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{28 - 39 y}{5}$ into the numerator.

$- 5 \frac{- (28 - 39 y)}{5} = - 5 \cdot 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3.2.1.1.1.2

Factor $5$ out of $- 5$ .

$5 (- 1) (\frac{- (28 - 39 y)}{5}) = - 5 \cdot 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3.2.1.1.1.3

Cancel the common factor.

$5 \cdot (- 1 \frac{- (28 - 39 y)}{5}) = - 5 \cdot 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3.2.1.1.1.4

Rewrite the expression.

$28 - 39 y = - 5 \cdot 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$28 - 39 y = - 5 \cdot 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (28 - 39 y) = - 5 \cdot 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3.2.1.1.2.2

Multiply $28 - 39 y$ by $1$ .

$28 - 39 y = - 5 \cdot 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$28 - 39 y = - 5 \cdot 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$28 - 39 y = - 5 \cdot 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$28 - 39 y = - 5 \cdot 10$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 5$ by $10$ .

$28 - 39 y = - 50$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$28 - 39 y = - 50$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$28 - 39 y = - 50$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3.3

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

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Step 3.3.1

Subtract $28$ from both sides of the equation.

$- 39 y = - 50 - 28$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3.3.2

Subtract $28$ from $- 50$ .

$- 39 y = - 78$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$- 39 y = - 78$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3.4

Divide each term in $- 39 y = - 78$ by $- 39$ and simplify.

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Step 3.4.1

Divide each term in $- 39 y = - 78$ by $- 39$ .

$\frac{- 39 y}{- 39} = \frac{- 78}{- 39}$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $- 39$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 39 y}{- 39} = \frac{- 78}{- 39}$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 78}{- 39}$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$y = \frac{- 78}{- 39}$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$y = \frac{- 78}{- 39}$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $- 78$ by $- 39$ .

$y = 2$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$y = 2$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$y = 2$

$x = - \frac{7}{10} + \frac{3 y}{5}$

$y = 2$

$x = - \frac{7}{10} + \frac{3 y}{5}$

Step 4

Replace all occurrences of $y$ with $2$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{7}{10} + \frac{3 y}{5}$ with $2$ .

$x = - \frac{7}{10} + \frac{3 (2)}{5}$

$y = 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{7}{10} + \frac{3 (2)}{5}$ .

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Step 4.2.1.1

Multiply $3$ by $2$ .

$x = - \frac{7}{10} + \frac{6}{5}$

$y = 2$

Step 4.2.1.2

To write $\frac{6}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = - \frac{7}{10} + \frac{6}{5} \cdot \frac{2}{2}$

$y = 2$

Step 4.2.1.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.3.1

Multiply $\frac{6}{5}$ by $\frac{2}{2}$ .

$x = - \frac{7}{10} + \frac{6 \cdot 2}{5 \cdot 2}$

$y = 2$

Step 4.2.1.3.2

Multiply $5$ by $2$ .

$x = - \frac{7}{10} + \frac{6 \cdot 2}{10}$

$y = 2$

$x = - \frac{7}{10} + \frac{6 \cdot 2}{10}$

$y = 2$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 7 + 6 \cdot 2}{10}$

$y = 2$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $6$ by $2$ .

$x = \frac{- 7 + 12}{10}$

$y = 2$

Step 4.2.1.5.2

Add $- 7$ and $12$ .

$x = \frac{5}{10}$

$y = 2$

$x = \frac{5}{10}$

$y = 2$

Step 4.2.1.6

Cancel the common factor of $5$ and $10$ .

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Step 4.2.1.6.1

Factor $5$ out of $5$ .

$x = \frac{5 (1)}{10}$

$y = 2$

Step 4.2.1.6.2

Cancel the common factors.

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Step 4.2.1.6.2.1

Factor $5$ out of $10$ .

$x = \frac{5 \cdot 1}{5 \cdot 2}$

$y = 2$

Step 4.2.1.6.2.2

Cancel the common factor.

$x = \frac{5 \cdot 1}{5 \cdot 2}$

$y = 2$

Step 4.2.1.6.2.3

Rewrite the expression.

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

$y = 2$

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

$y = 2$

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

$y = 2$

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

$y = 2$

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

$y = 2$

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

$y = 2$

Step 5

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

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

Step 6

The result can be shown in multiple forms.

Point Form:

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

Equation Form:

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

Step 7