Pre-Algebra Examples

$3 x - 5 y = 0$ , $5 x + 3 y = 15$

Step 1

Solve for $x$ in $3 x - 5 y = 0$ .

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Step 1.1

Add $5 y$ to both sides of the equation.

$3 x = 5 y$

$5 x + 3 y = 15$

Step 1.2

Divide each term in $3 x = 5 y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 5 y$ by $3$ .

$\frac{3 x}{3} = \frac{5 y}{3}$

$5 x + 3 y = 15$

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.

$\frac{3 x}{3} = \frac{5 y}{3}$

$5 x + 3 y = 15$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5 y}{3}$

$5 x + 3 y = 15$

$x = \frac{5 y}{3}$

$5 x + 3 y = 15$

$x = \frac{5 y}{3}$

$5 x + 3 y = 15$

$x = \frac{5 y}{3}$

$5 x + 3 y = 15$

$x = \frac{5 y}{3}$

$5 x + 3 y = 15$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $5 x + 3 y = 15$ with $\frac{5 y}{3}$ .

$5 (\frac{5 y}{3}) + 3 y = 15$

$x = \frac{5 y}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 (\frac{5 y}{3}) + 3 y$ .

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

$\frac{5 (5 y)}{3} + 3 y = 15$

$x = \frac{5 y}{3}$

Step 2.2.1.1.2

Multiply $5$ by $5$ .

$\frac{25 y}{3} + 3 y = 15$

$x = \frac{5 y}{3}$

$\frac{25 y}{3} + 3 y = 15$

$x = \frac{5 y}{3}$

Step 2.2.1.2

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

$\frac{25 y}{3} + 3 y \cdot \frac{3}{3} = 15$

$x = \frac{5 y}{3}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$\frac{25 y}{3} + \frac{3 y \cdot 3}{3} = 15$

$x = \frac{5 y}{3}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{25 y + 3 y \cdot 3}{3} = 15$

$x = \frac{5 y}{3}$

$\frac{25 y + 3 y \cdot 3}{3} = 15$

$x = \frac{5 y}{3}$

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Factor $y$ out of $25 y + 3 y \cdot 3$ .

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Step 2.2.1.4.1.1

Factor $y$ out of $25 y$ .

$\frac{y \cdot 25 + 3 y \cdot 3}{3} = 15$

$x = \frac{5 y}{3}$

Step 2.2.1.4.1.2

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

$\frac{y \cdot 25 + y (3 \cdot 3)}{3} = 15$

$x = \frac{5 y}{3}$

Step 2.2.1.4.1.3

Factor $y$ out of $y \cdot 25 + y (3 \cdot 3)$ .

$\frac{y (25 + 3 \cdot 3)}{3} = 15$

$x = \frac{5 y}{3}$

$\frac{y (25 + 3 \cdot 3)}{3} = 15$

$x = \frac{5 y}{3}$

Step 2.2.1.4.2

Multiply $3$ by $3$ .

$\frac{y (25 + 9)}{3} = 15$

$x = \frac{5 y}{3}$

Step 2.2.1.4.3

Add $25$ and $9$ .

$\frac{y \cdot 34}{3} = 15$

$x = \frac{5 y}{3}$

$\frac{y \cdot 34}{3} = 15$

$x = \frac{5 y}{3}$

Step 2.2.1.5

Move $34$ to the left of $y$ .

$\frac{34 y}{3} = 15$

$x = \frac{5 y}{3}$

$\frac{34 y}{3} = 15$

$x = \frac{5 y}{3}$

$\frac{34 y}{3} = 15$

$x = \frac{5 y}{3}$

$\frac{34 y}{3} = 15$

$x = \frac{5 y}{3}$

Step 3

Solve for $y$ in $\frac{34 y}{3} = 15$ .

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Step 3.1

Multiply both sides of the equation by $\frac{3}{34}$ .

$\frac{3}{34} \cdot \frac{34 y}{3} = \frac{3}{34} \cdot 15$

$x = \frac{5 y}{3}$

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 $\frac{3}{34} \cdot \frac{34 y}{3}$ .

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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.

$\frac{3}{34} \cdot \frac{34 y}{3} = \frac{3}{34} \cdot 15$

$x = \frac{5 y}{3}$

Step 3.2.1.1.1.2

Rewrite the expression.

$\frac{1}{34} \cdot (34 y) = \frac{3}{34} \cdot 15$

$x = \frac{5 y}{3}$

$\frac{1}{34} \cdot (34 y) = \frac{3}{34} \cdot 15$

$x = \frac{5 y}{3}$

Step 3.2.1.1.2

Cancel the common factor of $34$ .

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Step 3.2.1.1.2.1

Factor $34$ out of $34 y$ .

$\frac{1}{34} \cdot (34 (y)) = \frac{3}{34} \cdot 15$

$x = \frac{5 y}{3}$

Step 3.2.1.1.2.2

Cancel the common factor.

$\frac{1}{34} \cdot (34 y) = \frac{3}{34} \cdot 15$

$x = \frac{5 y}{3}$

Step 3.2.1.1.2.3

Rewrite the expression.

$y = \frac{3}{34} \cdot 15$

$x = \frac{5 y}{3}$

$y = \frac{3}{34} \cdot 15$

$x = \frac{5 y}{3}$

$y = \frac{3}{34} \cdot 15$

$x = \frac{5 y}{3}$

$y = \frac{3}{34} \cdot 15$

$x = \frac{5 y}{3}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $\frac{3}{34} \cdot 15$ .

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Step 3.2.2.1.1

Combine $\frac{3}{34}$ and $15$ .

$y = \frac{3 \cdot 15}{34}$

$x = \frac{5 y}{3}$

Step 3.2.2.1.2

Multiply $3$ by $15$ .

$y = \frac{45}{34}$

$x = \frac{5 y}{3}$

$y = \frac{45}{34}$

$x = \frac{5 y}{3}$

$y = \frac{45}{34}$

$x = \frac{5 y}{3}$

$y = \frac{45}{34}$

$x = \frac{5 y}{3}$

$y = \frac{45}{34}$

$x = \frac{5 y}{3}$

Step 4

Replace all occurrences of $y$ with $\frac{45}{34}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{5 y}{3}$ with $\frac{45}{34}$ .

$x = \frac{5 (\frac{45}{34})}{3}$

$y = \frac{45}{34}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{5 (\frac{45}{34})}{3}$ .

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Step 4.2.1.1

Combine $5$ and $\frac{45}{34}$ .

$x = \frac{\frac{5 \cdot 45}{34}}{3}$

$y = \frac{45}{34}$

Step 4.2.1.2

Multiply $5$ by $45$ .

$x = \frac{\frac{225}{34}}{3}$

$y = \frac{45}{34}$

Step 4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{225}{34} \cdot \frac{1}{3}$

$y = \frac{45}{34}$

Step 4.2.1.4

Cancel the common factor of $3$ .

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Step 4.2.1.4.1

Factor $3$ out of $225$ .

$x = \frac{3 (75)}{34} \cdot \frac{1}{3}$

$y = \frac{45}{34}$

Step 4.2.1.4.2

Cancel the common factor.

$x = \frac{3 \cdot 75}{34} \cdot \frac{1}{3}$

$y = \frac{45}{34}$

Step 4.2.1.4.3

Rewrite the expression.

$x = \frac{75}{34}$

$y = \frac{45}{34}$

$x = \frac{75}{34}$

$y = \frac{45}{34}$

$x = \frac{75}{34}$

$y = \frac{45}{34}$

$x = \frac{75}{34}$

$y = \frac{45}{34}$

$x = \frac{75}{34}$

$y = \frac{45}{34}$

Step 5

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

$(\frac{75}{34}, \frac{45}{34})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{75}{34}, \frac{45}{34})$

Equation Form:

$x = \frac{75}{34}, y = \frac{45}{34}$

Step 7