Precalculus Examples

$4 x^{2} + y = 3$ , $- x - y = 11$ , $(\frac{- 3}{2}, \frac{- 31}{3})$

Step 1

Simplify $4 {(\frac{- 3}{2})}^{2} + \frac{- 31}{3}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Move the negative in front of the fraction.

$4 {(- \frac{3}{2})}^{2} + \frac{- 31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.2.1

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

$4 ({(- 1)}^{2} {(\frac{3}{2})}^{2}) + \frac{- 31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.1.2.2

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

$4 ({(- 1)}^{2} (\frac{3^{2}}{2^{2}})) + \frac{- 31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

$4 ({(- 1)}^{2} (\frac{3^{2}}{2^{2}})) + \frac{- 31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.1.3

Raise $- 1$ to the power of $2$ .

$4 (1 (\frac{3^{2}}{2^{2}})) + \frac{- 31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.1.4

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

$4 (\frac{3^{2}}{2^{2}}) + \frac{- 31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.1.5

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

$4 (\frac{9}{2^{2}}) + \frac{- 31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.1.6

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

$4 (\frac{9}{4}) + \frac{- 31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.1.7

Cancel the common factor of $4$ .

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Step 1.1.7.1

Cancel the common factor.

$4 (\frac{9}{4}) + \frac{- 31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.1.7.2

Rewrite the expression.

$9 + \frac{- 31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

$9 + \frac{- 31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.1.8

Move the negative in front of the fraction.

$9 - \frac{31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

$9 - \frac{31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$9 \cdot \frac{3}{3} - \frac{31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.3

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

$\frac{9 \cdot 3}{3} - \frac{31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.4

Combine the numerators over the common denominator.

$\frac{9 \cdot 3 - 31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $9$ by $3$ .

$\frac{27 - 31}{3} = 3$

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.5.2

Subtract $31$ from $27$ .

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

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

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

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 1.6

Move the negative in front of the fraction.

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

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

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

$- (\frac{- 3}{2}) - (\frac{- 31}{3}) = 11$

Step 2

Since $- \frac{4}{3} \neq 3$ , there are no solutions.

No solution

Step 3

Simplify $- \frac{- 3}{2} - \frac{- 31}{3}$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Move the negative in front of the fraction.

$\frac{3}{2} - \frac{- 31}{3} = 11$

No solution

Step 3.1.2

Multiply $- - \frac{3}{2}$ .

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Step 3.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{3}{2}) - \frac{- 31}{3} = 11$

No solution

Step 3.1.2.2

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

$\frac{3}{2} - \frac{- 31}{3} = 11$

No solution

$\frac{3}{2} - \frac{- 31}{3} = 11$

No solution

Step 3.1.3

Move the negative in front of the fraction.

$\frac{3}{2} + \frac{31}{3} = 11$

No solution

Step 3.1.4

Multiply $- - \frac{31}{3}$ .

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Step 3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{3}{2} + 1 (\frac{31}{3}) = 11$

No solution

Step 3.1.4.2

Multiply $\frac{31}{3}$ by $1$ .

$\frac{3}{2} + \frac{31}{3} = 11$

No solution

$\frac{3}{2} + \frac{31}{3} = 11$

No solution

$\frac{3}{2} + \frac{31}{3} = 11$

No solution

Step 3.2

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3}{2} \cdot \frac{3}{3} + \frac{31}{3} = 11$

No solution

Step 3.3

To write $\frac{31}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2} \cdot \frac{3}{3} + \frac{31}{3} \cdot \frac{2}{2} = 11$

No solution

Step 3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

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

$\frac{3 \cdot 3}{2 \cdot 3} + \frac{31}{3} \cdot \frac{2}{2} = 11$

No solution

Step 3.4.2

Multiply $2$ by $3$ .

$\frac{3 \cdot 3}{6} + \frac{31}{3} \cdot \frac{2}{2} = 11$

No solution

Step 3.4.3

Multiply $\frac{31}{3}$ by $\frac{2}{2}$ .

$\frac{3 \cdot 3}{6} + \frac{31 \cdot 2}{3 \cdot 2} = 11$

No solution

Step 3.4.4

Multiply $3$ by $2$ .

$\frac{3 \cdot 3}{6} + \frac{31 \cdot 2}{6} = 11$

No solution

$\frac{3 \cdot 3}{6} + \frac{31 \cdot 2}{6} = 11$

No solution

Step 3.5

Combine the numerators over the common denominator.

$\frac{3 \cdot 3 + 31 \cdot 2}{6} = 11$

No solution

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $3$ by $3$ .

$\frac{9 + 31 \cdot 2}{6} = 11$

No solution

Step 3.6.2

Multiply $31$ by $2$ .

$\frac{9 + 62}{6} = 11$

No solution

Step 3.6.3

Add $9$ and $62$ .

$\frac{71}{6} = 11$

No solution

$\frac{71}{6} = 11$

No solution

Step 4

Since $\frac{71}{6} \neq 11$ , there are no solutions.

No solution

Step 5

The ordered pair is not a solution to the system of equations.

$(\frac{- 3}{2}, \frac{- 31}{3})$ is not a solution