Precalculus Examples

$y = 20 x - 16$ , $y = 12 x^{2} + 3 x - 10$

Step 1

Eliminate the equal sides of each equation and combine.

$20 x - 16 = 12 x^{2} + 3 x - 10$

Step 2

Solve $20 x - 16 = 12 x^{2} + 3 x - 10$ 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.

$12 x^{2} + 3 x - 10 = 20 x - 16$

Step 2.2

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

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Step 2.2.1

Subtract $20 x$ from both sides of the equation.

$12 x^{2} + 3 x - 10 - 20 x = - 16$

Step 2.2.2

Subtract $20 x$ from $3 x$ .

$12 x^{2} - 17 x - 10 = - 16$

Step 2.3

Add $16$ to both sides of the equation.

$12 x^{2} - 17 x - 10 + 16 = 0$

Step 2.4

Add $- 10$ and $16$ .

$12 x^{2} - 17 x + 6 = 0$

Step 2.5

Factor by grouping.

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Step 2.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 12 \cdot 6 = 72$ and whose sum is $b = - 17$ .

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Step 2.5.1.1

Factor $- 17$ out of $- 17 x$ .

$12 x^{2} - 17 x + 6 = 0$

Step 2.5.1.2

Rewrite $- 17$ as $- 8$ plus $- 9$

$12 x^{2} + (- 8 - 9) x + 6 = 0$

Step 2.5.1.3

Apply the distributive property.

$12 x^{2} - 8 x - 9 x + 6 = 0$

Step 2.5.2

Factor out the greatest common factor from each group.

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Step 2.5.2.1

Group the first two terms and the last two terms.

$(12 x^{2} - 8 x) - 9 x + 6 = 0$

Step 2.5.2.2

Factor out the greatest common factor (GCF) from each group.

$4 x (3 x - 2) - 3 (3 x - 2) = 0$

Step 2.5.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$(3 x - 2) (4 x - 3) = 0$

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 $0$ .

$3 x - 2 = 0$

$4 x - 3 = 0 + y = 12 x^{2} + 3 x - 10$

Step 2.7

Set $3 x - 2$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

Step 2.7.2

Solve $3 x - 2 = 0$ for $x$ .

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Step 2.7.2.1

Add $2$ to both sides of the equation.

$3 x = 2$

Step 2.7.2.2

Divide each term in $3 x = 2$ by $3$ and simplify.

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Step 2.7.2.2.1

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Step 2.7.2.2.2

Simplify the left side.

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Step 2.7.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.7.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Step 2.7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Step 2.8

Set $4 x - 3$ equal to $0$ and solve for $x$ .

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Step 2.8.1

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

Step 2.8.2

Solve $4 x - 3 = 0$ for $x$ .

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Step 2.8.2.1

Add $3$ to both sides of the equation.

$4 x = 3$

Step 2.8.2.2

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

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Step 2.8.2.2.1

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Step 2.8.2.2.2

Simplify the left side.

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Step 2.8.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.8.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Step 2.8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 2.9

The final solution is all the values that make $(3 x - 2) (4 x - 3) = 0$ true.

$x = \frac{2}{3}, \frac{3}{4}$

Step 3

Evaluate $y$ when $x = \frac{2}{3}$ .

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Step 3.1

Substitute $\frac{2}{3}$ for $x$ .

$y = 12 {(\frac{2}{3})}^{2} + 3 (\frac{2}{3}) - 10$

Step 3.2

Simplify $12 {(\frac{2}{3})}^{2} + 3 (\frac{2}{3}) - 10$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Apply the product rule to $\frac{2}{3}$ .

$y = 12 \frac{2^{2}}{3^{2}} + 3 (\frac{2}{3}) - 10$

Step 3.2.1.2

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

$y = 12 \frac{4}{3^{2}} + 3 (\frac{2}{3}) - 10$

Step 3.2.1.3

Raise $3$ to the power of $2$ .

$y = 12 (\frac{4}{9}) + 3 (\frac{2}{3}) - 10$

Step 3.2.1.4

Cancel the common factor of $3$ .

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Step 3.2.1.4.1

Factor $3$ out of $12$ .

$y = 3 (4) \frac{4}{9} + 3 (\frac{2}{3}) - 10$

Step 3.2.1.4.2

Factor $3$ out of $9$ .

$y = 3 \cdot 4 \frac{4}{3 \cdot 3} + 3 (\frac{2}{3}) - 10$

Step 3.2.1.4.3

Cancel the common factor.

$y = 3 \cdot 4 \frac{4}{3 \cdot 3} + 3 (\frac{2}{3}) - 10$

Step 3.2.1.4.4

Rewrite the expression.

$y = 4 (\frac{4}{3}) + 3 (\frac{2}{3}) - 10$

Step 3.2.1.5

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

$y = \frac{4 \cdot 4}{3} + 3 (\frac{2}{3}) - 10$

Step 3.2.1.6

Multiply $4$ by $4$ .

$y = \frac{16}{3} + 3 (\frac{2}{3}) - 10$

Step 3.2.1.7

Cancel the common factor of $3$ .

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Step 3.2.1.7.1

Cancel the common factor.

$y = \frac{16}{3} + 3 (\frac{2}{3}) - 10$

Step 3.2.1.7.2

Rewrite the expression.

$y = \frac{16}{3} + 2 - 10$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Write $2$ as a fraction with denominator $1$ .

$y = \frac{16}{3} + \frac{2}{1} - 10$

Step 3.2.2.2

Multiply $\frac{2}{1}$ by $\frac{3}{3}$ .

$y = \frac{16}{3} + \frac{2}{1} \cdot \frac{3}{3} - 10$

Step 3.2.2.3

Multiply $\frac{2}{1}$ by $\frac{3}{3}$ .

$y = \frac{16}{3} + \frac{2 \cdot 3}{3} - 10$

Step 3.2.2.4

Write $- 10$ as a fraction with denominator $1$ .

$y = \frac{16}{3} + \frac{2 \cdot 3}{3} + \frac{- 10}{1}$

Step 3.2.2.5

Multiply $\frac{- 10}{1}$ by $\frac{3}{3}$ .

$y = \frac{16}{3} + \frac{2 \cdot 3}{3} + \frac{- 10}{1} \cdot \frac{3}{3}$

Step 3.2.2.6

Multiply $\frac{- 10}{1}$ by $\frac{3}{3}$ .

$y = \frac{16}{3} + \frac{2 \cdot 3}{3} + \frac{- 10 \cdot 3}{3}$

Step 3.2.3

Combine the numerators over the common denominator.

$y = \frac{16 + 2 \cdot 3 - 10 \cdot 3}{3}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply $2$ by $3$ .

$y = \frac{16 + 6 - 10 \cdot 3}{3}$

Step 3.2.4.2

Multiply $- 10$ by $3$ .

$y = \frac{16 + 6 - 30}{3}$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Add $16$ and $6$ .

$y = \frac{22 - 30}{3}$

Step 3.2.5.2

Subtract $30$ from $22$ .

$y = \frac{- 8}{3}$

Step 3.2.5.3

Move the negative in front of the fraction.

$y = - \frac{8}{3}$

Step 4

Evaluate $y$ when $x = \frac{3}{4}$ .

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Step 4.1

Substitute $\frac{3}{4}$ for $x$ .

$y = 12 {(\frac{3}{4})}^{2} + 3 (\frac{3}{4}) - 10$

Step 4.2

Simplify $12 {(\frac{3}{4})}^{2} + 3 (\frac{3}{4}) - 10$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Apply the product rule to $\frac{3}{4}$ .

$y = 12 \frac{3^{2}}{4^{2}} + 3 (\frac{3}{4}) - 10$

Step 4.2.1.2

Raise $3$ to the power of $2$ .

$y = 12 \frac{9}{4^{2}} + 3 (\frac{3}{4}) - 10$

Step 4.2.1.3

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

$y = 12 (\frac{9}{16}) + 3 (\frac{3}{4}) - 10$

Step 4.2.1.4

Cancel the common factor of $4$ .

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Step 4.2.1.4.1

Factor $4$ out of $12$ .

$y = 4 (3) \frac{9}{16} + 3 (\frac{3}{4}) - 10$

Step 4.2.1.4.2

Factor $4$ out of $16$ .

$y = 4 \cdot 3 \frac{9}{4 \cdot 4} + 3 (\frac{3}{4}) - 10$

Step 4.2.1.4.3

Cancel the common factor.

$y = 4 \cdot 3 \frac{9}{4 \cdot 4} + 3 (\frac{3}{4}) - 10$

Step 4.2.1.4.4

Rewrite the expression.

$y = 3 (\frac{9}{4}) + 3 (\frac{3}{4}) - 10$

Step 4.2.1.5

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

$y = \frac{3 \cdot 9}{4} + 3 (\frac{3}{4}) - 10$

Step 4.2.1.6

Multiply $3$ by $9$ .

$y = \frac{27}{4} + 3 (\frac{3}{4}) - 10$

Step 4.2.1.7

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

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Step 4.2.1.7.1

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

$y = \frac{27}{4} + \frac{3 \cdot 3}{4} - 10$

Step 4.2.1.7.2

Multiply $3$ by $3$ .

$y = \frac{27}{4} + \frac{9}{4} - 10$

Step 4.2.2

Combine fractions.

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Step 4.2.2.1

Combine the numerators over the common denominator.

$y = - 10 + \frac{27 + 9}{4}$

Step 4.2.2.2

Simplify the expression.

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Step 4.2.2.2.1

Add $27$ and $9$ .

$y = - 10 + \frac{36}{4}$

Step 4.2.2.2.2

Divide $36$ by $4$ .

$y = - 10 + 9$

Step 4.2.2.2.3

Add $- 10$ and $9$ .

$y = - 1$

Step 5

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

$(\frac{2}{3}, - \frac{8}{3})$

$(\frac{3}{4}, - 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{2}{3}, - \frac{8}{3}), (\frac{3}{4}, - 1)$

Equation Form:

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

$x = \frac{3}{4}, y = - 1$

Step 7