Finite Math Examples

$\frac{7}{4} x - \frac{5}{2} y = 2$ , $\frac{1}{4} x + \frac{7}{2} y = 2$

Step 1

Solve for $x$ in $\frac{7}{4} x - \frac{5}{2} y = 2$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$\frac{7 x}{4} - \frac{5}{2} y = 2$

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

Step 1.1.2

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

$\frac{7 x}{4} - \frac{y \cdot 5}{2} = 2$

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

Step 1.1.3

Move $5$ to the left of $y$ .

$\frac{7 x}{4} - \frac{5 y}{2} = 2$

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

$\frac{7 x}{4} - \frac{5 y}{2} = 2$

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

Step 1.2

Add $\frac{5 y}{2}$ to both sides of the equation.

$\frac{7 x}{4} = 2 + \frac{5 y}{2}$

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

Step 1.3

Multiply both sides of the equation by $\frac{4}{7}$ .

$\frac{4}{7} \cdot \frac{7 x}{4} = \frac{4}{7} \cdot (2 + \frac{5 y}{2})$

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

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 $\frac{4}{7} \cdot \frac{7 x}{4}$ .

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Step 1.4.1.1.1

Cancel the common factor of $4$ .

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Step 1.4.1.1.1.1

Cancel the common factor.

$\frac{4}{7} \cdot \frac{7 x}{4} = \frac{4}{7} \cdot (2 + \frac{5 y}{2})$

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

Step 1.4.1.1.1.2

Rewrite the expression.

$\frac{1}{7} \cdot (7 x) = \frac{4}{7} \cdot (2 + \frac{5 y}{2})$

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

$\frac{1}{7} \cdot (7 x) = \frac{4}{7} \cdot (2 + \frac{5 y}{2})$

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

Step 1.4.1.1.2

Cancel the common factor of $7$ .

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Step 1.4.1.1.2.1

Factor $7$ out of $7 x$ .

$\frac{1}{7} \cdot (7 (x)) = \frac{4}{7} \cdot (2 + \frac{5 y}{2})$

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

Step 1.4.1.1.2.2

Cancel the common factor.

$\frac{1}{7} \cdot (7 x) = \frac{4}{7} \cdot (2 + \frac{5 y}{2})$

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

Step 1.4.1.1.2.3

Rewrite the expression.

$x = \frac{4}{7} \cdot (2 + \frac{5 y}{2})$

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

$x = \frac{4}{7} \cdot (2 + \frac{5 y}{2})$

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

$x = \frac{4}{7} \cdot (2 + \frac{5 y}{2})$

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

$x = \frac{4}{7} \cdot (2 + \frac{5 y}{2})$

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

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify $\frac{4}{7} (2 + \frac{5 y}{2})$ .

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Step 1.4.2.1.1

Apply the distributive property.

$x = \frac{4}{7} \cdot 2 + \frac{4}{7} \cdot \frac{5 y}{2}$

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

Step 1.4.2.1.2

Multiply $\frac{4}{7} \cdot 2$ .

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Step 1.4.2.1.2.1

Combine $\frac{4}{7}$ and $2$ .

$x = \frac{4 \cdot 2}{7} + \frac{4}{7} \cdot \frac{5 y}{2}$

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

Step 1.4.2.1.2.2

Multiply $4$ by $2$ .

$x = \frac{8}{7} + \frac{4}{7} \cdot \frac{5 y}{2}$

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

$x = \frac{8}{7} + \frac{4}{7} \cdot \frac{5 y}{2}$

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

Step 1.4.2.1.3

Cancel the common factor of $2$ .

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Step 1.4.2.1.3.1

Factor $2$ out of $4$ .

$x = \frac{8}{7} + \frac{2 (2)}{7} \cdot \frac{5 y}{2}$

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

Step 1.4.2.1.3.2

Cancel the common factor.

$x = \frac{8}{7} + \frac{2 \cdot 2}{7} \cdot \frac{5 y}{2}$

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

Step 1.4.2.1.3.3

Rewrite the expression.

$x = \frac{8}{7} + \frac{2}{7} \cdot (5 y)$

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

$x = \frac{8}{7} + \frac{2}{7} \cdot (5 y)$

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

Step 1.4.2.1.4

Combine $5$ and $\frac{2}{7}$ .

$x = \frac{8}{7} + \frac{5 \cdot 2}{7} y$

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

Step 1.4.2.1.5

Multiply $5$ by $2$ .

$x = \frac{8}{7} + \frac{10}{7} y$

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

Step 1.4.2.1.6

Combine $\frac{10}{7}$ and $y$ .

$x = \frac{8}{7} + \frac{10 y}{7}$

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

$x = \frac{8}{7} + \frac{10 y}{7}$

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

$x = \frac{8}{7} + \frac{10 y}{7}$

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

$x = \frac{8}{7} + \frac{10 y}{7}$

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

Step 1.5

Reorder $\frac{8}{7}$ and $\frac{10 y}{7}$ .

$x = \frac{10 y}{7} + \frac{8}{7}$

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

$x = \frac{10 y}{7} + \frac{8}{7}$

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

Step 2

Replace all occurrences of $x$ with $\frac{10 y}{7} + \frac{8}{7}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $\frac{1}{4} x + \frac{7}{2} y = 2$ with $\frac{10 y}{7} + \frac{8}{7}$ .

$\frac{1}{4} \cdot (\frac{10 y}{7} + \frac{8}{7}) + \frac{7}{2} y = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{1}{4} \cdot (\frac{10 y}{7} + \frac{8}{7}) + \frac{7}{2} y$ .

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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.

$\frac{1}{4} \cdot \frac{10 y}{7} + \frac{1}{4} \cdot \frac{8}{7} + \frac{7}{2} y = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Factor $2$ out of $4$ .

$\frac{1}{2 (2)} \cdot \frac{10 y}{7} + \frac{1}{4} \cdot \frac{8}{7} + \frac{7}{2} y = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.1.2.2

Factor $2$ out of $10 y$ .

$\frac{1}{2 (2)} \cdot \frac{2 (5 y)}{7} + \frac{1}{4} \cdot \frac{8}{7} + \frac{7}{2} y = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.1.2.3

Cancel the common factor.

$\frac{1}{2 \cdot 2} \cdot \frac{2 (5 y)}{7} + \frac{1}{4} \cdot \frac{8}{7} + \frac{7}{2} y = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.1.2.4

Rewrite the expression.

$\frac{1}{2} \cdot \frac{5 y}{7} + \frac{1}{4} \cdot \frac{8}{7} + \frac{7}{2} y = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{1}{2} \cdot \frac{5 y}{7} + \frac{1}{4} \cdot \frac{8}{7} + \frac{7}{2} y = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.1.3

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

$\frac{5 y}{2 \cdot 7} + \frac{1}{4} \cdot \frac{8}{7} + \frac{7}{2} y = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.1.4

Multiply $2$ by $7$ .

$\frac{5 y}{14} + \frac{1}{4} \cdot \frac{8}{7} + \frac{7}{2} y = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.1.5

Cancel the common factor of $4$ .

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Step 2.2.1.1.5.1

Factor $4$ out of $8$ .

$\frac{5 y}{14} + \frac{1}{4} \cdot \frac{4 (2)}{7} + \frac{7}{2} y = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.1.5.2

Cancel the common factor.

$\frac{5 y}{14} + \frac{1}{4} \cdot \frac{4 \cdot 2}{7} + \frac{7}{2} y = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.1.5.3

Rewrite the expression.

$\frac{5 y}{14} + \frac{2}{7} + \frac{7}{2} y = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{5 y}{14} + \frac{2}{7} + \frac{7}{2} y = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.1.6

Combine $\frac{7}{2}$ and $y$ .

$\frac{5 y}{14} + \frac{2}{7} + \frac{7 y}{2} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{5 y}{14} + \frac{2}{7} + \frac{7 y}{2} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.2

To write $\frac{7 y}{2}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{5 y}{14} + \frac{7 y}{2} \cdot \frac{7}{7} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.3

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.3.1

Multiply $\frac{7 y}{2}$ by $\frac{7}{7}$ .

$\frac{5 y}{14} + \frac{7 y \cdot 7}{2 \cdot 7} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.3.2

Multiply $2$ by $7$ .

$\frac{5 y}{14} + \frac{7 y \cdot 7}{14} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{5 y}{14} + \frac{7 y \cdot 7}{14} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{5 y + 7 y \cdot 7}{14} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.5

Simplify each term.

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Step 2.2.1.5.1

Simplify the numerator.

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Step 2.2.1.5.1.1

Factor $y$ out of $5 y + 7 y \cdot 7$ .

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Step 2.2.1.5.1.1.1

Factor $y$ out of $5 y$ .

$\frac{y \cdot 5 + 7 y \cdot 7}{14} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.5.1.1.2

Factor $y$ out of $7 y \cdot 7$ .

$\frac{y \cdot 5 + y (7 \cdot 7)}{14} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.5.1.1.3

Factor $y$ out of $y \cdot 5 + y (7 \cdot 7)$ .

$\frac{y (5 + 7 \cdot 7)}{14} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{y (5 + 7 \cdot 7)}{14} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.5.1.2

Multiply $7$ by $7$ .

$\frac{y (5 + 49)}{14} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.5.1.3

Add $5$ and $49$ .

$\frac{y \cdot 54}{14} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{y \cdot 54}{14} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.5.2

Cancel the common factor of $54$ and $14$ .

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Step 2.2.1.5.2.1

Factor $2$ out of $y \cdot 54$ .

$\frac{2 (y \cdot 27)}{14} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.5.2.2

Cancel the common factors.

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Step 2.2.1.5.2.2.1

Factor $2$ out of $14$ .

$\frac{2 (y \cdot 27)}{2 (7)} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.5.2.2.2

Cancel the common factor.

$\frac{2 (y \cdot 27)}{2 \cdot 7} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.5.2.2.3

Rewrite the expression.

$\frac{y \cdot 27}{7} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{y \cdot 27}{7} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{y \cdot 27}{7} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 2.2.1.5.3

Move $27$ to the left of $y$ .

$\frac{27 y}{7} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{27 y}{7} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{27 y}{7} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{27 y}{7} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{27 y}{7} + \frac{2}{7} = 2$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3

Solve for $y$ in $\frac{27 y}{7} + \frac{2}{7} = 2$ .

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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 $\frac{2}{7}$ from both sides of the equation.

$\frac{27 y}{7} = 2 - \frac{2}{7}$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3.1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{27 y}{7} = 2 \cdot \frac{7}{7} - \frac{2}{7}$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3.1.3

Combine $2$ and $\frac{7}{7}$ .

$\frac{27 y}{7} = \frac{2 \cdot 7}{7} - \frac{2}{7}$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{27 y}{7} = \frac{2 \cdot 7 - 2}{7}$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $2$ by $7$ .

$\frac{27 y}{7} = \frac{14 - 2}{7}$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3.1.5.2

Subtract $2$ from $14$ .

$\frac{27 y}{7} = \frac{12}{7}$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{27 y}{7} = \frac{12}{7}$

$x = \frac{10 y}{7} + \frac{8}{7}$

$\frac{27 y}{7} = \frac{12}{7}$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$27 y = 12$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3.3

Divide each term in $27 y = 12$ by $27$ and simplify.

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Step 3.3.1

Divide each term in $27 y = 12$ by $27$ .

$\frac{27 y}{27} = \frac{12}{27}$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $27$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{27 y}{27} = \frac{12}{27}$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{12}{27}$

$x = \frac{10 y}{7} + \frac{8}{7}$

$y = \frac{12}{27}$

$x = \frac{10 y}{7} + \frac{8}{7}$

$y = \frac{12}{27}$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $12$ and $27$ .

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Step 3.3.3.1.1

Factor $3$ out of $12$ .

$y = \frac{3 (4)}{27}$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $3$ out of $27$ .

$y = \frac{3 \cdot 4}{3 \cdot 9}$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3.3.3.1.2.2

Cancel the common factor.

$y = \frac{3 \cdot 4}{3 \cdot 9}$

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 3.3.3.1.2.3

Rewrite the expression.

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

$x = \frac{10 y}{7} + \frac{8}{7}$

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

$x = \frac{10 y}{7} + \frac{8}{7}$

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

$x = \frac{10 y}{7} + \frac{8}{7}$

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

$x = \frac{10 y}{7} + \frac{8}{7}$

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

$x = \frac{10 y}{7} + \frac{8}{7}$

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

$x = \frac{10 y}{7} + \frac{8}{7}$

Step 4

Replace all occurrences of $y$ with $\frac{4}{9}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{10 y}{7} + \frac{8}{7}$ with $\frac{4}{9}$ .

$x = \frac{10 (\frac{4}{9})}{7} + \frac{8}{7}$

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

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{10 (\frac{4}{9})}{7} + \frac{8}{7}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{10 (\frac{4}{9}) + 8}{7}$

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

Step 4.2.1.2

Multiply $10 (\frac{4}{9})$ .

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Step 4.2.1.2.1

Combine $10$ and $\frac{4}{9}$ .

$x = \frac{\frac{10 \cdot 4}{9} + 8}{7}$

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

Step 4.2.1.2.2

Multiply $10$ by $4$ .

$x = \frac{\frac{40}{9} + 8}{7}$

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

$x = \frac{\frac{40}{9} + 8}{7}$

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

Step 4.2.1.3

To write $8$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$x = \frac{\frac{40}{9} + 8 \cdot \frac{9}{9}}{7}$

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

Step 4.2.1.4

Combine $8$ and $\frac{9}{9}$ .

$x = \frac{\frac{40}{9} + \frac{8 \cdot 9}{9}}{7}$

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

Step 4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{40 + 8 \cdot 9}{9}}{7}$

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

Step 4.2.1.6

Simplify the numerator.

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Step 4.2.1.6.1

Multiply $8$ by $9$ .

$x = \frac{\frac{40 + 72}{9}}{7}$

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

Step 4.2.1.6.2

Add $40$ and $72$ .

$x = \frac{\frac{112}{9}}{7}$

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

$x = \frac{\frac{112}{9}}{7}$

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

Step 4.2.1.7

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{112}{9} \cdot \frac{1}{7}$

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

Step 4.2.1.8

Cancel the common factor of $7$ .

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Step 4.2.1.8.1

Factor $7$ out of $112$ .

$x = \frac{7 (16)}{9} \cdot \frac{1}{7}$

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

Step 4.2.1.8.2

Cancel the common factor.

$x = \frac{7 \cdot 16}{9} \cdot \frac{1}{7}$

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

Step 4.2.1.8.3

Rewrite the expression.

$x = \frac{16}{9}$

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

$x = \frac{16}{9}$

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

$x = \frac{16}{9}$

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

$x = \frac{16}{9}$

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

$x = \frac{16}{9}$

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

Step 5

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

$(\frac{16}{9}, \frac{4}{9})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{16}{9}, \frac{4}{9})$

Equation Form:

$x = \frac{16}{9}, y = \frac{4}{9}$

Step 7