Finite Math Examples

$3 x - y = 0$ , $5 x + 2 y = 22$

Step 1

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

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Step 1.1

Add $y$ to both sides of the equation.

$3 x = y$

$5 x + 2 y = 22$

Step 1.2

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

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Step 1.2.1

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

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

$5 x + 2 y = 22$

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.

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

$5 x + 2 y = 22$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$5 x + 2 y = 22$

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

$5 x + 2 y = 22$

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

$5 x + 2 y = 22$

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

$5 x + 2 y = 22$

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

$5 x + 2 y = 22$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $5 x + 2 y = 22$ with $\frac{y}{3}$ .

$5 (\frac{y}{3}) + 2 y = 22$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

$\frac{5 y}{3} + 2 y = 22$

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

Step 2.2.1.2

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5 y}{3} + 2 y \cdot \frac{3}{3} = 22$

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

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$\frac{5 y}{3} + \frac{2 y \cdot 3}{3} = 22$

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

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{5 y + 2 y \cdot 3}{3} = 22$

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

$\frac{5 y + 2 y \cdot 3}{3} = 22$

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

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Factor $y$ out of $5 y + 2 y \cdot 3$ .

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Step 2.2.1.4.1.1

Factor $y$ out of $5 y$ .

$\frac{y \cdot 5 + 2 y \cdot 3}{3} = 22$

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

Step 2.2.1.4.1.2

Factor $y$ out of $2 y \cdot 3$ .

$\frac{y \cdot 5 + y (2 \cdot 3)}{3} = 22$

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

Step 2.2.1.4.1.3

Factor $y$ out of $y \cdot 5 + y (2 \cdot 3)$ .

$\frac{y (5 + 2 \cdot 3)}{3} = 22$

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

$\frac{y (5 + 2 \cdot 3)}{3} = 22$

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

Step 2.2.1.4.2

Multiply $2$ by $3$ .

$\frac{y (5 + 6)}{3} = 22$

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

Step 2.2.1.4.3

Add $5$ and $6$ .

$\frac{y \cdot 11}{3} = 22$

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

$\frac{y \cdot 11}{3} = 22$

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

Step 2.2.1.5

Move $11$ to the left of $y$ .

$\frac{11 y}{3} = 22$

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

$\frac{11 y}{3} = 22$

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

$\frac{11 y}{3} = 22$

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

$\frac{11 y}{3} = 22$

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

Step 3

Solve for $y$ in $\frac{11 y}{3} = 22$ .

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Step 3.1

Multiply both sides of the equation by $\frac{3}{11}$ .

$\frac{3}{11} \cdot \frac{11 y}{3} = \frac{3}{11} \cdot 22$

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

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 $\frac{3}{11} \cdot \frac{11 y}{3}$ .

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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.

$\frac{3}{11} \cdot \frac{11 y}{3} = \frac{3}{11} \cdot 22$

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

Step 3.2.1.1.1.2

Rewrite the expression.

$\frac{1}{11} \cdot (11 y) = \frac{3}{11} \cdot 22$

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

$\frac{1}{11} \cdot (11 y) = \frac{3}{11} \cdot 22$

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

Step 3.2.1.1.2

Cancel the common factor of $11$ .

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Step 3.2.1.1.2.1

Factor $11$ out of $11 y$ .

$\frac{1}{11} \cdot (11 (y)) = \frac{3}{11} \cdot 22$

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

Step 3.2.1.1.2.2

Cancel the common factor.

$\frac{1}{11} \cdot (11 y) = \frac{3}{11} \cdot 22$

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

Step 3.2.1.1.2.3

Rewrite the expression.

$y = \frac{3}{11} \cdot 22$

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

$y = \frac{3}{11} \cdot 22$

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

$y = \frac{3}{11} \cdot 22$

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

$y = \frac{3}{11} \cdot 22$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{3}{11} \cdot 22$ .

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Step 3.2.2.1.1

Cancel the common factor of $11$ .

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Step 3.2.2.1.1.1

Factor $11$ out of $22$ .

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

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

Step 3.2.2.1.1.2

Cancel the common factor.

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

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

Step 3.2.2.1.1.3

Rewrite the expression.

$y = 3 \cdot 2$

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

$y = 3 \cdot 2$

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

Step 3.2.2.1.2

Multiply $3$ by $2$ .

$y = 6$

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

$y = 6$

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

$y = 6$

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

$y = 6$

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

$y = 6$

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

Step 4

Replace all occurrences of $y$ with $6$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{y}{3}$ with $6$ .

$x = \frac{6}{3}$

$y = 6$

Step 4.2

Simplify the right side.

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Step 4.2.1

Divide $6$ by $3$ .

$x = 2$

$y = 6$

$x = 2$

$y = 6$

$x = 2$

$y = 6$

Step 5

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

$(2, 6)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, 6)$

Equation Form:

$x = 2, y = 6$

Step 7