Finite Math Examples

$y = 3 x - 6$ , $y = x^{2} - x - 6$

Step 1

Eliminate the equal sides of each equation and combine.

$3 x - 6 = x^{2} - x - 6$

Step 2

Solve $3 x - 6 = x^{2} - x - 6$ for $x$ .

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Step 2.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - x - 6 = 3 x - 6$

Step 2.2

Move all terms containing $x$ to the left side of the equation.

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Step 2.2.1

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

$x^{2} - x - 6 - 3 x = - 6$

Step 2.2.2

Subtract $3 x$ from $- x$ .

$x^{2} - 4 x - 6 = - 6$

Step 2.3

Add $6$ to both sides of the equation.

$x^{2} - 4 x - 6 + 6 = 0$

Step 2.4

Combine the opposite terms in $x^{2} - 4 x - 6 + 6$ .

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Step 2.4.1

Add $- 6$ and $6$ .

$x^{2} - 4 x + 0 = 0$

Step 2.4.2

Add $x^{2} - 4 x$ and $0$ .

$x^{2} - 4 x = 0$

Step 2.5

Factor $x$ out of $x^{2} - 4 x$ .

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Step 2.5.1

Factor $x$ out of $x^{2}$ .

$x \cdot x - 4 x = 0$

Step 2.5.2

Factor $x$ out of $- 4 x$ .

$x \cdot x + x \cdot - 4 = 0$

Step 2.5.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$x (x - 4) = 0$

Step 2.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 4 = 0 + y = x^{2} - x - 6$

Step 2.7

Set $x$ equal to $0$ .

$x = 0$

Step 2.8

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.8.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.8.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.9

The final solution is all the values that make $x (x - 4) = 0$ true.

$x = 0, 4$

Step 3

Evaluate $y$ when $x = 0$ .

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Step 3.1

Substitute $0$ for $x$ .

$y = {(0)}^{2} - (0) - 6$

Step 3.2

Substitute $0$ for $x$ in $y = {(0)}^{2} - (0) - 6$ and solve for $y$ .

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Step 3.2.1

Remove parentheses.

$y = 0^{2} - (0) - 6$

Step 3.2.2

Simplify $0^{2} - (0) - 6$ .

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Raising $0$ to any positive power yields $0$ .

$y = 0 - (0) - 6$

Step 3.2.2.1.2

Multiply $- 1$ by $0$ .

$y = 0 + 0 - 6$

Step 3.2.2.2

Simplify by adding and subtracting.

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Step 3.2.2.2.1

Add $0$ and $0$ .

$y = 0 - 6$

Step 3.2.2.2.2

Subtract $6$ from $0$ .

$y = - 6$

Step 4

Evaluate $y$ when $x = 4$ .

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Step 4.1

Substitute $4$ for $x$ .

$y = {(4)}^{2} - (4) - 6$

Step 4.2

Substitute $4$ for $x$ in $y = {(4)}^{2} - (4) - 6$ and solve for $y$ .

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Step 4.2.1

Remove parentheses.

$y = 4^{2} - (4) - 6$

Step 4.2.2

Simplify $4^{2} - (4) - 6$ .

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Raise $4$ to the power of $2$ .

$y = 16 - (4) - 6$

Step 4.2.2.1.2

Multiply $- 1$ by $4$ .

$y = 16 - 4 - 6$

Step 4.2.2.2

Simplify by subtracting numbers.

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Step 4.2.2.2.1

Subtract $4$ from $16$ .

$y = 12 - 6$

Step 4.2.2.2.2

Subtract $6$ from $12$ .

$y = 6$

Step 5

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

$(0, - 6)$

$(4, 6)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(0, - 6), (4, 6)$

Equation Form:

$x = 0, y = - 6$

$x = 4, y = 6$

Step 7