Finite Math Examples

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

Step 1

Subtract $3 x$ from both sides of the equation.

$y = 7 - 3 x$

$x - y = 2$

Step 2

Replace all occurrences of $y$ with $7 - 3 x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x - y = 2$ with $7 - 3 x$ .

$x - (7 - 3 x) = 2$

$y = 7 - 3 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x - (7 - 3 x)$ .

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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.

$x - 1 \cdot 7 - (- 3 x) = 2$

$y = 7 - 3 x$

Step 2.2.1.1.2

Multiply $- 1$ by $7$ .

$x - 7 - (- 3 x) = 2$

$y = 7 - 3 x$

Step 2.2.1.1.3

Multiply $- 3$ by $- 1$ .

$x - 7 + 3 x = 2$

$y = 7 - 3 x$

$x - 7 + 3 x = 2$

$y = 7 - 3 x$

Step 2.2.1.2

Add $x$ and $3 x$ .

$4 x - 7 = 2$

$y = 7 - 3 x$

$4 x - 7 = 2$

$y = 7 - 3 x$

$4 x - 7 = 2$

$y = 7 - 3 x$

$4 x - 7 = 2$

$y = 7 - 3 x$

Step 3

Solve for $x$ in $4 x - 7 = 2$ .

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Step 3.1

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

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Step 3.1.1

Add $7$ to both sides of the equation.

$4 x = 2 + 7$

$y = 7 - 3 x$

Step 3.1.2

Add $2$ and $7$ .

$4 x = 9$

$y = 7 - 3 x$

$4 x = 9$

$y = 7 - 3 x$

Step 3.2

Divide each term in $4 x = 9$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 x = 9$ by $4$ .

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

$y = 7 - 3 x$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

$y = 7 - 3 x$

Step 3.2.2.1.2

Divide $x$ by $1$ .

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

$y = 7 - 3 x$

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

$y = 7 - 3 x$

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

$y = 7 - 3 x$

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

$y = 7 - 3 x$

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

$y = 7 - 3 x$

Step 4

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

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Step 4.1

Replace all occurrences of $x$ in $y = 7 - 3 x$ with $\frac{9}{4}$ .

$y = 7 - 3 (\frac{9}{4})$

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

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $7 - 3 (\frac{9}{4})$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Multiply $- 3 (\frac{9}{4})$ .

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Step 4.2.1.1.1.1

Combine $- 3$ and $\frac{9}{4}$ .

$y = 7 + \frac{- 3 \cdot 9}{4}$

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

Step 4.2.1.1.1.2

Multiply $- 3$ by $9$ .

$y = 7 + \frac{- 27}{4}$

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

$y = 7 + \frac{- 27}{4}$

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

Step 4.2.1.1.2

Move the negative in front of the fraction.

$y = 7 - \frac{27}{4}$

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

$y = 7 - \frac{27}{4}$

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

Step 4.2.1.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = 7 \cdot \frac{4}{4} - \frac{27}{4}$

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

Step 4.2.1.3

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

$y = \frac{7 \cdot 4}{4} - \frac{27}{4}$

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

Step 4.2.1.4

Combine the numerators over the common denominator.

$y = \frac{7 \cdot 4 - 27}{4}$

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

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $7$ by $4$ .

$y = \frac{28 - 27}{4}$

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

Step 4.2.1.5.2

Subtract $27$ from $28$ .

$y = \frac{1}{4}$

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

$y = \frac{1}{4}$

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

$y = \frac{1}{4}$

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

$y = \frac{1}{4}$

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

$y = \frac{1}{4}$

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

Step 5

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

$(\frac{9}{4}, \frac{1}{4})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{9}{4}, \frac{1}{4})$

Equation Form:

$x = \frac{9}{4}, y = \frac{1}{4}$

Step 7