Finite Math Examples

$4 x - 3 y = 45$ , $5 x + 2 y = 16$

Step 1

Solve for $x$ in $4 x - 3 y = 45$ .

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Step 1.1

Add $3 y$ to both sides of the equation.

$4 x = 45 + 3 y$

$5 x + 2 y = 16$

Step 1.2

Divide each term in $4 x = 45 + 3 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = 45 + 3 y$ by $4$ .

$\frac{4 x}{4} = \frac{45}{4} + \frac{3 y}{4}$

$5 x + 2 y = 16$

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.

$\frac{4 x}{4} = \frac{45}{4} + \frac{3 y}{4}$

$5 x + 2 y = 16$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{45}{4} + \frac{3 y}{4}$

$5 x + 2 y = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

$5 x + 2 y = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

$5 x + 2 y = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

$5 x + 2 y = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

$5 x + 2 y = 16$

Step 2

Replace all occurrences of $x$ with $\frac{45}{4} + \frac{3 y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $5 x + 2 y = 16$ with $\frac{45}{4} + \frac{3 y}{4}$ .

$5 (\frac{45}{4} + \frac{3 y}{4}) + 2 y = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 (\frac{45}{4} + \frac{3 y}{4}) + 2 y$ .

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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.

$5 (\frac{45}{4}) + 5 (\frac{3 y}{4}) + 2 y = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 2.2.1.1.2

Multiply $5 (\frac{45}{4})$ .

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Step 2.2.1.1.2.1

Combine $5$ and $\frac{45}{4}$ .

$\frac{5 \cdot 45}{4} + 5 (\frac{3 y}{4}) + 2 y = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 2.2.1.1.2.2

Multiply $5$ by $45$ .

$\frac{225}{4} + 5 (\frac{3 y}{4}) + 2 y = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

$\frac{225}{4} + 5 (\frac{3 y}{4}) + 2 y = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 2.2.1.1.3

Multiply $5 \frac{3 y}{4}$ .

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Step 2.2.1.1.3.1

Combine $5$ and $\frac{3 y}{4}$ .

$\frac{225}{4} + \frac{5 (3 y)}{4} + 2 y = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 2.2.1.1.3.2

Multiply $3$ by $5$ .

$\frac{225}{4} + \frac{15 y}{4} + 2 y = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

$\frac{225}{4} + \frac{15 y}{4} + 2 y = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

$\frac{225}{4} + \frac{15 y}{4} + 2 y = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 2.2.1.2

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{225}{4} + \frac{15 y}{4} + 2 y \cdot \frac{4}{4} = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 2.2.1.3

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

$\frac{225}{4} + \frac{15 y}{4} + \frac{2 y \cdot 4}{4} = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{225}{4} + \frac{15 y + 2 y \cdot 4}{4} = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{225 + 15 y + 2 y \cdot 4}{4} = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 2.2.1.6

Multiply $4$ by $2$ .

$\frac{225 + 15 y + 8 y}{4} = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 2.2.1.7

Add $15 y$ and $8 y$ .

$\frac{225 + 23 y}{4} = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

$\frac{225 + 23 y}{4} = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

$\frac{225 + 23 y}{4} = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

$\frac{225 + 23 y}{4} = 16$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 3

Solve for $y$ in $\frac{225 + 23 y}{4} = 16$ .

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Step 3.1

Multiply both sides by $4$ .

$\frac{225 + 23 y}{4} \cdot 4 = 16 \cdot 4$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{225 + 23 y}{4} \cdot 4$ .

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Step 3.2.1.1.1

Cancel the common factor of $4$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{225 + 23 y}{4} \cdot 4 = 16 \cdot 4$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 3.2.1.1.1.2

Rewrite the expression.

$225 + 23 y = 16 \cdot 4$

$x = \frac{45}{4} + \frac{3 y}{4}$

$225 + 23 y = 16 \cdot 4$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 3.2.1.1.2

Reorder $225$ and $23 y$ .

$23 y + 225 = 16 \cdot 4$

$x = \frac{45}{4} + \frac{3 y}{4}$

$23 y + 225 = 16 \cdot 4$

$x = \frac{45}{4} + \frac{3 y}{4}$

$23 y + 225 = 16 \cdot 4$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $16$ by $4$ .

$23 y + 225 = 64$

$x = \frac{45}{4} + \frac{3 y}{4}$

$23 y + 225 = 64$

$x = \frac{45}{4} + \frac{3 y}{4}$

$23 y + 225 = 64$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.1.1

Subtract $225$ from both sides of the equation.

$23 y = 64 - 225$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 3.3.1.2

Subtract $225$ from $64$ .

$23 y = - 161$

$x = \frac{45}{4} + \frac{3 y}{4}$

$23 y = - 161$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 3.3.2

Divide each term in $23 y = - 161$ by $23$ and simplify.

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Step 3.3.2.1

Divide each term in $23 y = - 161$ by $23$ .

$\frac{23 y}{23} = \frac{- 161}{23}$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $23$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{23 y}{23} = \frac{- 161}{23}$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 161}{23}$

$x = \frac{45}{4} + \frac{3 y}{4}$

$y = \frac{- 161}{23}$

$x = \frac{45}{4} + \frac{3 y}{4}$

$y = \frac{- 161}{23}$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $- 161$ by $23$ .

$y = - 7$

$x = \frac{45}{4} + \frac{3 y}{4}$

$y = - 7$

$x = \frac{45}{4} + \frac{3 y}{4}$

$y = - 7$

$x = \frac{45}{4} + \frac{3 y}{4}$

$y = - 7$

$x = \frac{45}{4} + \frac{3 y}{4}$

$y = - 7$

$x = \frac{45}{4} + \frac{3 y}{4}$

Step 4

Replace all occurrences of $y$ with $- 7$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{45}{4} + \frac{3 y}{4}$ with $- 7$ .

$x = \frac{45}{4} + \frac{3 (- 7)}{4}$

$y = - 7$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{45}{4} + \frac{3 (- 7)}{4}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{45 + 3 (- 7)}{4}$

$y = - 7$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $3$ by $- 7$ .

$x = \frac{45 - 21}{4}$

$y = - 7$

Step 4.2.1.2.2

Subtract $21$ from $45$ .

$x = \frac{24}{4}$

$y = - 7$

Step 4.2.1.2.3

Divide $24$ by $4$ .

$x = 6$

$y = - 7$

$x = 6$

$y = - 7$

$x = 6$

$y = - 7$

$x = 6$

$y = - 7$

$x = 6$

$y = - 7$

Step 5

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

$(6, - 7)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(6, - 7)$

Equation Form:

$x = 6, y = - 7$

Step 7