Finite Math Examples

$y = x^{2}$ , $3 x = y + 2$

Step 1

Replace all occurrences of $y$ with $x^{2}$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $3 x = y + 2$ with $x^{2}$ .

$3 x = (x^{2}) + 2$

$y = x^{2}$

Step 1.2

Simplify the right side.

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Step 1.2.1

Remove parentheses.

$3 x = x^{2} + 2$

$y = x^{2}$

$3 x = x^{2} + 2$

$y = x^{2}$

$3 x = x^{2} + 2$

$y = x^{2}$

Step 2

Solve for $x$ in $3 x = x^{2} + 2$ .

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Step 2.1

Subtract $x^{2}$ from both sides of the equation.

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

$y = x^{2}$

Step 2.2

Subtract $2$ from both sides of the equation.

$3 x - x^{2} - 2 = 0$

$y = x^{2}$

Step 2.3

Factor the left side of the equation.

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Step 2.3.1

Factor $- 1$ out of $3 x - x^{2} - 2$ .

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Step 2.3.1.1

Reorder $3 x$ and $- x^{2}$ .

$- x^{2} + 3 x - 2 = 0$

$y = x^{2}$

Step 2.3.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 3 x - 2 = 0$

$y = x^{2}$

Step 2.3.1.3

Factor $- 1$ out of $3 x$ .

$- (x^{2}) - (- 3 x) - 2 = 0$

$y = x^{2}$

Step 2.3.1.4

Rewrite $- 2$ as $- 1 (2)$ .

$- (x^{2}) - (- 3 x) - 1 \cdot 2 = 0$

$y = x^{2}$

Step 2.3.1.5

Factor $- 1$ out of $- (x^{2}) - (- 3 x)$ .

$- (x^{2} - 3 x) - 1 \cdot 2 = 0$

$y = x^{2}$

Step 2.3.1.6

Factor $- 1$ out of $- (x^{2} - 3 x) - 1 (2)$ .

$- (x^{2} - 3 x + 2) = 0$

$y = x^{2}$

$- (x^{2} - 3 x + 2) = 0$

$y = x^{2}$

Step 2.3.2

Factor.

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Step 2.3.2.1

Factor $x^{2} - 3 x + 2$ using the AC method.

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Step 2.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

$y = x^{2}$

Step 2.3.2.1.2

Write the factored form using these integers.

$- ((x - 2) (x - 1)) = 0$

$y = x^{2}$

$- ((x - 2) (x - 1)) = 0$

$y = x^{2}$

Step 2.3.2.2

Remove unnecessary parentheses.

$- (x - 2) (x - 1) = 0$

$y = x^{2}$

$- (x - 2) (x - 1) = 0$

$y = x^{2}$

$- (x - 2) (x - 1) = 0$

$y = x^{2}$

Step 2.4

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

$x - 2 = 0$

$x - 1 = 0$

$y = x^{2}$

Step 2.5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

$y = x^{2}$

Step 2.5.2

Add $2$ to both sides of the equation.

$x = 2$

$y = x^{2}$

$x = 2$

$y = x^{2}$

Step 2.6

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

$y = x^{2}$

Step 2.6.2

Add $1$ to both sides of the equation.

$x = 1$

$y = x^{2}$

$x = 1$

$y = x^{2}$

Step 2.7

The final solution is all the values that make $- (x - 2) (x - 1) = 0$ true.

$x = 2, 1$

$y = x^{2}$

$x = 2, 1$

$y = x^{2}$

Step 3

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

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Step 3.1

Replace all occurrences of $x$ in $y = x^{2}$ with $2$ .

$y = {(2)}^{2}$

$x = 2$

Step 3.2

Simplify the right side.

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Step 3.2.1

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

$y = 4$

$x = 2$

$y = 4$

$x = 2$

$y = 4$

$x = 2$

Step 4

Replace all occurrences of $x$ with $1$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = x^{2}$ with $1$ .

$y = {(1)}^{2}$

$x = 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

One to any power is one.

$y = 1$

$x = 1$

$y = 1$

$x = 1$

$y = 1$

$x = 1$

Step 5

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

$(2, 4)$

$(1, 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, 4), (1, 1)$

Equation Form:

$x = 2, y = 4$

$x = 1, y = 1$

Step 7