Finite Math Examples

$4 x - 2 y = 36$ , $3 x + 7 y = - 7$

Step 1

Solve for $x$ in $4 x - 2 y = 36$ .

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Step 1.1

Add $2 y$ to both sides of the equation.

$4 x = 36 + 2 y$

$3 x + 7 y = - 7$

Step 1.2

Divide each term in $4 x = 36 + 2 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = 36 + 2 y$ by $4$ .

$\frac{4 x}{4} = \frac{36}{4} + \frac{2 y}{4}$

$3 x + 7 y = - 7$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{36}{4} + \frac{2 y}{4}$

$3 x + 7 y = - 7$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{36}{4} + \frac{2 y}{4}$

$3 x + 7 y = - 7$

$x = \frac{36}{4} + \frac{2 y}{4}$

$3 x + 7 y = - 7$

$x = \frac{36}{4} + \frac{2 y}{4}$

$3 x + 7 y = - 7$

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

Divide $36$ by $4$ .

$x = 9 + \frac{2 y}{4}$

$3 x + 7 y = - 7$

Step 1.2.3.1.2

Cancel the common factor of $2$ and $4$ .

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Step 1.2.3.1.2.1

Factor $2$ out of $2 y$ .

$x = 9 + \frac{2 (y)}{4}$

$3 x + 7 y = - 7$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $4$ .

$x = 9 + \frac{2 y}{2 \cdot 2}$

$3 x + 7 y = - 7$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = 9 + \frac{2 y}{2 \cdot 2}$

$3 x + 7 y = - 7$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = 9 + \frac{y}{2}$

$3 x + 7 y = - 7$

$x = 9 + \frac{y}{2}$

$3 x + 7 y = - 7$

$x = 9 + \frac{y}{2}$

$3 x + 7 y = - 7$

$x = 9 + \frac{y}{2}$

$3 x + 7 y = - 7$

$x = 9 + \frac{y}{2}$

$3 x + 7 y = - 7$

$x = 9 + \frac{y}{2}$

$3 x + 7 y = - 7$

$x = 9 + \frac{y}{2}$

$3 x + 7 y = - 7$

Step 2

Replace all occurrences of $x$ with $9 + \frac{y}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x + 7 y = - 7$ with $9 + \frac{y}{2}$ .

$3 (9 + \frac{y}{2}) + 7 y = - 7$

$x = 9 + \frac{y}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (9 + \frac{y}{2}) + 7 y$ .

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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.

$3 \cdot 9 + 3 (\frac{y}{2}) + 7 y = - 7$

$x = 9 + \frac{y}{2}$

Step 2.2.1.1.2

Multiply $3$ by $9$ .

$27 + 3 (\frac{y}{2}) + 7 y = - 7$

$x = 9 + \frac{y}{2}$

Step 2.2.1.1.3

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

$27 + \frac{3 y}{2} + 7 y = - 7$

$x = 9 + \frac{y}{2}$

$27 + \frac{3 y}{2} + 7 y = - 7$

$x = 9 + \frac{y}{2}$

Step 2.2.1.2

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

$27 + \frac{3 y}{2} + 7 y \cdot \frac{2}{2} = - 7$

$x = 9 + \frac{y}{2}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$27 + \frac{3 y}{2} + \frac{7 y \cdot 2}{2} = - 7$

$x = 9 + \frac{y}{2}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$27 + \frac{3 y + 7 y \cdot 2}{2} = - 7$

$x = 9 + \frac{y}{2}$

$27 + \frac{3 y + 7 y \cdot 2}{2} = - 7$

$x = 9 + \frac{y}{2}$

Step 2.2.1.4

Simplify each term.

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Step 2.2.1.4.1

Simplify the numerator.

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Step 2.2.1.4.1.1

Factor $y$ out of $3 y + 7 y \cdot 2$ .

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Step 2.2.1.4.1.1.1

Factor $y$ out of $3 y$ .

$27 + \frac{y \cdot 3 + 7 y \cdot 2}{2} = - 7$

$x = 9 + \frac{y}{2}$

Step 2.2.1.4.1.1.2

Factor $y$ out of $7 y \cdot 2$ .

$27 + \frac{y \cdot 3 + y (7 \cdot 2)}{2} = - 7$

$x = 9 + \frac{y}{2}$

Step 2.2.1.4.1.1.3

Factor $y$ out of $y \cdot 3 + y (7 \cdot 2)$ .

$27 + \frac{y (3 + 7 \cdot 2)}{2} = - 7$

$x = 9 + \frac{y}{2}$

$27 + \frac{y (3 + 7 \cdot 2)}{2} = - 7$

$x = 9 + \frac{y}{2}$

Step 2.2.1.4.1.2

Multiply $7$ by $2$ .

$27 + \frac{y (3 + 14)}{2} = - 7$

$x = 9 + \frac{y}{2}$

Step 2.2.1.4.1.3

Add $3$ and $14$ .

$27 + \frac{y \cdot 17}{2} = - 7$

$x = 9 + \frac{y}{2}$

$27 + \frac{y \cdot 17}{2} = - 7$

$x = 9 + \frac{y}{2}$

Step 2.2.1.4.2

Move $17$ to the left of $y$ .

$27 + \frac{17 y}{2} = - 7$

$x = 9 + \frac{y}{2}$

$27 + \frac{17 y}{2} = - 7$

$x = 9 + \frac{y}{2}$

$27 + \frac{17 y}{2} = - 7$

$x = 9 + \frac{y}{2}$

$27 + \frac{17 y}{2} = - 7$

$x = 9 + \frac{y}{2}$

$27 + \frac{17 y}{2} = - 7$

$x = 9 + \frac{y}{2}$

Step 3

Solve for $y$ in $27 + \frac{17 y}{2} = - 7$ .

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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 $27$ from both sides of the equation.

$\frac{17 y}{2} = - 7 - 27$

$x = 9 + \frac{y}{2}$

Step 3.1.2

Subtract $27$ from $- 7$ .

$\frac{17 y}{2} = - 34$

$x = 9 + \frac{y}{2}$

$\frac{17 y}{2} = - 34$

$x = 9 + \frac{y}{2}$

Step 3.2

Multiply both sides of the equation by $\frac{2}{17}$ .

$\frac{2}{17} \cdot \frac{17 y}{2} = \frac{2}{17} \cdot - 34$

$x = 9 + \frac{y}{2}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{2}{17} \cdot \frac{17 y}{2}$ .

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Step 3.3.1.1.1

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{2}{17} \cdot \frac{17 y}{2} = \frac{2}{17} \cdot - 34$

$x = 9 + \frac{y}{2}$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{17} \cdot (17 y) = \frac{2}{17} \cdot - 34$

$x = 9 + \frac{y}{2}$

$\frac{1}{17} \cdot (17 y) = \frac{2}{17} \cdot - 34$

$x = 9 + \frac{y}{2}$

Step 3.3.1.1.2

Cancel the common factor of $17$ .

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Step 3.3.1.1.2.1

Factor $17$ out of $17 y$ .

$\frac{1}{17} \cdot (17 (y)) = \frac{2}{17} \cdot - 34$

$x = 9 + \frac{y}{2}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{17} \cdot (17 y) = \frac{2}{17} \cdot - 34$

$x = 9 + \frac{y}{2}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{2}{17} \cdot - 34$

$x = 9 + \frac{y}{2}$

$y = \frac{2}{17} \cdot - 34$

$x = 9 + \frac{y}{2}$

$y = \frac{2}{17} \cdot - 34$

$x = 9 + \frac{y}{2}$

$y = \frac{2}{17} \cdot - 34$

$x = 9 + \frac{y}{2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{2}{17} \cdot - 34$ .

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Step 3.3.2.1.1

Cancel the common factor of $17$ .

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Step 3.3.2.1.1.1

Factor $17$ out of $- 34$ .

$y = \frac{2}{17} \cdot (17 (- 2))$

$x = 9 + \frac{y}{2}$

Step 3.3.2.1.1.2

Cancel the common factor.

$y = \frac{2}{17} \cdot (17 \cdot - 2)$

$x = 9 + \frac{y}{2}$

Step 3.3.2.1.1.3

Rewrite the expression.

$y = 2 \cdot - 2$

$x = 9 + \frac{y}{2}$

$y = 2 \cdot - 2$

$x = 9 + \frac{y}{2}$

Step 3.3.2.1.2

Multiply $2$ by $- 2$ .

$y = - 4$

$x = 9 + \frac{y}{2}$

$y = - 4$

$x = 9 + \frac{y}{2}$

$y = - 4$

$x = 9 + \frac{y}{2}$

$y = - 4$

$x = 9 + \frac{y}{2}$

$y = - 4$

$x = 9 + \frac{y}{2}$

Step 4

Replace all occurrences of $y$ with $- 4$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 9 + \frac{y}{2}$ with $- 4$ .

$x = 9 + \frac{- 4}{2}$

$y = - 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $9 + \frac{- 4}{2}$ .

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Step 4.2.1.1

Divide $- 4$ by $2$ .

$x = 9 - 2$

$y = - 4$

Step 4.2.1.2

Subtract $2$ from $9$ .

$x = 7$

$y = - 4$

$x = 7$

$y = - 4$

$x = 7$

$y = - 4$

$x = 7$

$y = - 4$

Step 5

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

$(7, - 4)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(7, - 4)$

Equation Form:

$x = 7, y = - 4$

Step 7