Finite Math Examples

$x + 2 y = 8$ , $\frac{1}{2} x + \frac{1}{2} y = 3$

Step 1

Subtract $2 y$ from both sides of the equation.

$x = 8 - 2 y$

$\frac{1}{2} x + \frac{1}{2} y = 3$

Step 2

Replace all occurrences of $x$ with $8 - 2 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $\frac{1}{2} x + \frac{1}{2} y = 3$ with $8 - 2 y$ .

$\frac{1}{2} \cdot (8 - 2 y) + \frac{1}{2} y = 3$

$x = 8 - 2 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{1}{2} \cdot (8 - 2 y) + \frac{1}{2} y$ .

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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.

$\frac{1}{2} \cdot 8 + \frac{1}{2} \cdot (- 2 y) + \frac{1}{2} y = 3$

$x = 8 - 2 y$

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 $8$ .

$\frac{1}{2} \cdot (2 (4)) + \frac{1}{2} \cdot (- 2 y) + \frac{1}{2} y = 3$

$x = 8 - 2 y$

Step 2.2.1.1.2.2

Cancel the common factor.

$\frac{1}{2} \cdot (2 \cdot 4) + \frac{1}{2} \cdot (- 2 y) + \frac{1}{2} y = 3$

$x = 8 - 2 y$

Step 2.2.1.1.2.3

Rewrite the expression.

$4 + \frac{1}{2} \cdot (- 2 y) + \frac{1}{2} y = 3$

$x = 8 - 2 y$

$4 + \frac{1}{2} \cdot (- 2 y) + \frac{1}{2} y = 3$

$x = 8 - 2 y$

Step 2.2.1.1.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.3.1

Factor $2$ out of $- 2 y$ .

$4 + \frac{1}{2} \cdot (2 (- y)) + \frac{1}{2} y = 3$

$x = 8 - 2 y$

Step 2.2.1.1.3.2

Cancel the common factor.

$4 + \frac{1}{2} \cdot (2 (- y)) + \frac{1}{2} y = 3$

$x = 8 - 2 y$

Step 2.2.1.1.3.3

Rewrite the expression.

$4 - y + \frac{1}{2} y = 3$

$x = 8 - 2 y$

$4 - y + \frac{1}{2} y = 3$

$x = 8 - 2 y$

Step 2.2.1.1.4

Combine $\frac{1}{2}$ and $y$ .

$4 - y + \frac{y}{2} = 3$

$x = 8 - 2 y$

$4 - y + \frac{y}{2} = 3$

$x = 8 - 2 y$

Step 2.2.1.2

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

$4 - y \cdot \frac{2}{2} + \frac{y}{2} = 3$

$x = 8 - 2 y$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$4 + \frac{- y \cdot 2}{2} + \frac{y}{2} = 3$

$x = 8 - 2 y$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$4 + \frac{- y \cdot 2 + y}{2} = 3$

$x = 8 - 2 y$

$4 + \frac{- y \cdot 2 + y}{2} = 3$

$x = 8 - 2 y$

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 $- y \cdot 2 + y$ .

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Step 2.2.1.4.1.1.1

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

$4 + \frac{y (- 1 \cdot 2) + y}{2} = 3$

$x = 8 - 2 y$

Step 2.2.1.4.1.1.2

Raise $y$ to the power of $1$ .

$4 + \frac{y (- 1 \cdot 2) + y}{2} = 3$

$x = 8 - 2 y$

Step 2.2.1.4.1.1.3

Factor $y$ out of $y^{1}$ .

$4 + \frac{y (- 1 \cdot 2) + y \cdot 1}{2} = 3$

$x = 8 - 2 y$

Step 2.2.1.4.1.1.4

Factor $y$ out of $y (- 1 \cdot 2) + y \cdot 1$ .

$4 + \frac{y (- 1 \cdot 2 + 1)}{2} = 3$

$x = 8 - 2 y$

$4 + \frac{y (- 1 \cdot 2 + 1)}{2} = 3$

$x = 8 - 2 y$

Step 2.2.1.4.1.2

Multiply $- 1$ by $2$ .

$4 + \frac{y (- 2 + 1)}{2} = 3$

$x = 8 - 2 y$

Step 2.2.1.4.1.3

Add $- 2$ and $1$ .

$4 + \frac{y \cdot - 1}{2} = 3$

$x = 8 - 2 y$

$4 + \frac{y \cdot - 1}{2} = 3$

$x = 8 - 2 y$

Step 2.2.1.4.2

Move $- 1$ to the left of $y$ .

$4 + \frac{- 1 \cdot y}{2} = 3$

$x = 8 - 2 y$

Step 2.2.1.4.3

Move the negative in front of the fraction.

$4 - \frac{y}{2} = 3$

$x = 8 - 2 y$

$4 - \frac{y}{2} = 3$

$x = 8 - 2 y$

$4 - \frac{y}{2} = 3$

$x = 8 - 2 y$

$4 - \frac{y}{2} = 3$

$x = 8 - 2 y$

$4 - \frac{y}{2} = 3$

$x = 8 - 2 y$

Step 3

Solve for $y$ in $4 - \frac{y}{2} = 3$ .

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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 $4$ from both sides of the equation.

$- \frac{y}{2} = 3 - 4$

$x = 8 - 2 y$

Step 3.1.2

Subtract $4$ from $3$ .

$- \frac{y}{2} = - 1$

$x = 8 - 2 y$

$- \frac{y}{2} = - 1$

$x = 8 - 2 y$

Step 3.2

Multiply both sides of the equation by $- 2$ .

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

$x = 8 - 2 y$

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 $- 2 (- \frac{y}{2})$ .

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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

Move the leading negative in $- \frac{y}{2}$ into the numerator.

$- 2 \frac{- y}{2} = - 2 \cdot - 1$

$x = 8 - 2 y$

Step 3.3.1.1.1.2

Factor $2$ out of $- 2$ .

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

$x = 8 - 2 y$

Step 3.3.1.1.1.3

Cancel the common factor.

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

$x = 8 - 2 y$

Step 3.3.1.1.1.4

Rewrite the expression.

$y = - 2 \cdot - 1$

$x = 8 - 2 y$

$y = - 2 \cdot - 1$

$x = 8 - 2 y$

Step 3.3.1.1.2

Multiply.

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Step 3.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 y = - 2 \cdot - 1$

$x = 8 - 2 y$

Step 3.3.1.1.2.2

Multiply $y$ by $1$ .

$y = - 2 \cdot - 1$

$x = 8 - 2 y$

$y = - 2 \cdot - 1$

$x = 8 - 2 y$

$y = - 2 \cdot - 1$

$x = 8 - 2 y$

$y = - 2 \cdot - 1$

$x = 8 - 2 y$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $- 2$ by $- 1$ .

$y = 2$

$x = 8 - 2 y$

$y = 2$

$x = 8 - 2 y$

$y = 2$

$x = 8 - 2 y$

$y = 2$

$x = 8 - 2 y$

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 $x = 8 - 2 y$ with $2$ .

$x = 8 - 2 \cdot 2$

$y = 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $8 - 2 \cdot 2$ .

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Step 4.2.1.1

Multiply $- 2$ by $2$ .

$x = 8 - 4$

$y = 2$

Step 4.2.1.2

Subtract $4$ from $8$ .

$x = 4$

$y = 2$

$x = 4$

$y = 2$

$x = 4$

$y = 2$

$x = 4$

$y = 2$

Step 5

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

$(4, 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(4, 2)$

Equation Form:

$x = 4, y = 2$

Step 7