Precalculus Examples

$3 x^{2} + 30 x - y = - 23$ , $- x^{2} + 6 x - y = - 3$

Step 1

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

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Step 1.1

Subtract $3 x^{2}$ from both sides of the equation.

$30 x - y = - 23 - 3 x^{2}$

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

Step 1.2

Subtract $30 x$ from both sides of the equation.

$- y = - 23 - 3 x^{2} - 30 x$

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

$- y = - 23 - 3 x^{2} - 30 x$

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

Step 2

Divide each term in $- y = - 23 - 3 x^{2} - 30 x$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- y = - 23 - 3 x^{2} - 30 x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 23}{- 1} + \frac{- 3 x^{2}}{- 1} + \frac{- 30 x}{- 1}$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 23}{- 1} + \frac{- 3 x^{2}}{- 1} + \frac{- 30 x}{- 1}$

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

Step 2.2.2

Divide $y$ by $1$ .

$y = \frac{- 23}{- 1} + \frac{- 3 x^{2}}{- 1} + \frac{- 30 x}{- 1}$

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

$y = \frac{- 23}{- 1} + \frac{- 3 x^{2}}{- 1} + \frac{- 30 x}{- 1}$

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Divide $- 23$ by $- 1$ .

$y = 23 + \frac{- 3 x^{2}}{- 1} + \frac{- 30 x}{- 1}$

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

Step 2.3.1.2

Move the negative one from the denominator of $\frac{- 3 x^{2}}{- 1}$ .

$y = 23 - 1 \cdot (- 3 x^{2}) + \frac{- 30 x}{- 1}$

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

Step 2.3.1.3

Rewrite $- 1 \cdot (- 3 x^{2})$ as $- (- 3 x^{2})$ .

$y = 23 - (- 3 x^{2}) + \frac{- 30 x}{- 1}$

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

Step 2.3.1.4

Multiply $- 3$ by $- 1$ .

$y = 23 + 3 x^{2} + \frac{- 30 x}{- 1}$

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

Step 2.3.1.5

Move the negative one from the denominator of $\frac{- 30 x}{- 1}$ .

$y = 23 + 3 x^{2} - 1 \cdot (- 30 x)$

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

Step 2.3.1.6

Rewrite $- 1 \cdot (- 30 x)$ as $- (- 30 x)$ .

$y = 23 + 3 x^{2} - (- 30 x)$

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

Step 2.3.1.7

Multiply $- 30$ by $- 1$ .

$y = 23 + 3 x^{2} + 30 x$

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

$y = 23 + 3 x^{2} + 30 x$

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

$y = 23 + 3 x^{2} + 30 x$

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

$y = 23 + 3 x^{2} + 30 x$

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

Step 3

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

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Step 3.1

Add $x^{2}$ to both sides of the equation.

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

$y = 23 + 3 x^{2} + 30 x$

Step 3.2

Subtract $6 x$ from both sides of the equation.

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

$y = 23 + 3 x^{2} + 30 x$

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

$y = 23 + 3 x^{2} + 30 x$

Step 4

Divide each term in $- y = - 3 + x^{2} - 6 x$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- y = - 3 + x^{2} - 6 x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 3}{- 1} + \frac{x^{2}}{- 1} + \frac{- 6 x}{- 1}$

$y = 23 + 3 x^{2} + 30 x$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 3}{- 1} + \frac{x^{2}}{- 1} + \frac{- 6 x}{- 1}$

$y = 23 + 3 x^{2} + 30 x$

Step 4.2.2

Divide $y$ by $1$ .

$y = \frac{- 3}{- 1} + \frac{x^{2}}{- 1} + \frac{- 6 x}{- 1}$

$y = 23 + 3 x^{2} + 30 x$

$y = \frac{- 3}{- 1} + \frac{x^{2}}{- 1} + \frac{- 6 x}{- 1}$

$y = 23 + 3 x^{2} + 30 x$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Divide $- 3$ by $- 1$ .

$y = 3 + \frac{x^{2}}{- 1} + \frac{- 6 x}{- 1}$

$y = 23 + 3 x^{2} + 30 x$

Step 4.3.1.2

Move the negative one from the denominator of $\frac{x^{2}}{- 1}$ .

$y = 3 - 1 \cdot x^{2} + \frac{- 6 x}{- 1}$

$y = 23 + 3 x^{2} + 30 x$

Step 4.3.1.3

Rewrite $- 1 \cdot x^{2}$ as $- x^{2}$ .

$y = 3 - x^{2} + \frac{- 6 x}{- 1}$

$y = 23 + 3 x^{2} + 30 x$

Step 4.3.1.4

Move the negative one from the denominator of $\frac{- 6 x}{- 1}$ .

$y = 3 - x^{2} - 1 \cdot (- 6 x)$

$y = 23 + 3 x^{2} + 30 x$

Step 4.3.1.5

Rewrite $- 1 \cdot (- 6 x)$ as $- (- 6 x)$ .

$y = 3 - x^{2} - (- 6 x)$

$y = 23 + 3 x^{2} + 30 x$

Step 4.3.1.6

Multiply $- 6$ by $- 1$ .

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

$y = 23 + 3 x^{2} + 30 x$

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

$y = 23 + 3 x^{2} + 30 x$

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

$y = 23 + 3 x^{2} + 30 x$

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

$y = 23 + 3 x^{2} + 30 x$

Step 5

Create a graph to locate the intersection of the equations. The intersection of the system of equations is the solution.

$(- 1, - 4)$

$(- 5, - 52)$

Step 6