Precalculus Examples

$y + x^{2} = 7 x$ , $y + 7 x = 49$

Step 1

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

$y = 7 x - x^{2}$

$y + 7 x = 49$

Step 2

Replace all occurrences of $y$ with $7 x - x^{2}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $y + 7 x = 49$ with $7 x - x^{2}$ .

$(7 x - x^{2}) + 7 x = 49$

$y = 7 x - x^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Add $7 x$ and $7 x$ .

$- x^{2} + 14 x = 49$

$y = 7 x - x^{2}$

$- x^{2} + 14 x = 49$

$y = 7 x - x^{2}$

$- x^{2} + 14 x = 49$

$y = 7 x - x^{2}$

Step 3

Solve for $x$ in $- x^{2} + 14 x = 49$ .

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Step 3.1

Subtract $49$ from both sides of the equation.

$- x^{2} + 14 x - 49 = 0$

$y = 7 x - x^{2}$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $- 1$ out of $- x^{2} + 14 x - 49$ .

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Step 3.2.1.1

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

$- (x^{2}) + 14 x - 49 = 0$

$y = 7 x - x^{2}$

Step 3.2.1.2

Factor $- 1$ out of $14 x$ .

$- (x^{2}) - (- 14 x) - 49 = 0$

$y = 7 x - x^{2}$

Step 3.2.1.3

Rewrite $- 49$ as $- 1 (49)$ .

$- (x^{2}) - (- 14 x) - 1 \cdot 49 = 0$

$y = 7 x - x^{2}$

Step 3.2.1.4

Factor $- 1$ out of $- (x^{2}) - (- 14 x)$ .

$- (x^{2} - 14 x) - 1 \cdot 49 = 0$

$y = 7 x - x^{2}$

Step 3.2.1.5

Factor $- 1$ out of $- (x^{2} - 14 x) - 1 (49)$ .

$- (x^{2} - 14 x + 49) = 0$

$y = 7 x - x^{2}$

$- (x^{2} - 14 x + 49) = 0$

$y = 7 x - x^{2}$

Step 3.2.2

Factor using the perfect square rule.

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Step 3.2.2.1

Rewrite $49$ as $7^{2}$ .

$- (x^{2} - 14 x + 7^{2}) = 0$

$y = 7 x - x^{2}$

Step 3.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 x = 2 \cdot x \cdot 7$

$y = 7 x - x^{2}$

Step 3.2.2.3

Rewrite the polynomial.

$- (x^{2} - 2 \cdot x \cdot 7 + 7^{2}) = 0$

$y = 7 x - x^{2}$

Step 3.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 7$ .

$- {(x - 7)}^{2} = 0$

$y = 7 x - x^{2}$

$- {(x - 7)}^{2} = 0$

$y = 7 x - x^{2}$

$- {(x - 7)}^{2} = 0$

$y = 7 x - x^{2}$

Step 3.3

Divide each term in $- {(x - 7)}^{2} = 0$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- {(x - 7)}^{2} = 0$ by $- 1$ .

$\frac{- {(x - 7)}^{2}}{- 1} = \frac{0}{- 1}$

$y = 7 x - x^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{{(x - 7)}^{2}}{1} = \frac{0}{- 1}$

$y = 7 x - x^{2}$

Step 3.3.2.2

Divide ${(x - 7)}^{2}$ by $1$ .

${(x - 7)}^{2} = \frac{0}{- 1}$

$y = 7 x - x^{2}$

${(x - 7)}^{2} = \frac{0}{- 1}$

$y = 7 x - x^{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0$ by $- 1$ .

${(x - 7)}^{2} = 0$

$y = 7 x - x^{2}$

${(x - 7)}^{2} = 0$

$y = 7 x - x^{2}$

${(x - 7)}^{2} = 0$

$y = 7 x - x^{2}$

Step 3.4

Set the $x - 7$ equal to $0$ .

$x - 7 = 0$

$y = 7 x - x^{2}$

Step 3.5

Add $7$ to both sides of the equation.

$x = 7$

$y = 7 x - x^{2}$

$x = 7$

$y = 7 x - x^{2}$

Step 4

Replace all occurrences of $x$ with $7$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = 7 x - x^{2}$ with $7$ .

$y = 7 (7) - {(7)}^{2}$

$x = 7$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $7 (7) - {(7)}^{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Multiply $7$ by $7$ .

$y = 49 - {(7)}^{2}$

$x = 7$

Step 4.2.1.1.2

Raise $7$ to the power of $2$ .

$y = 49 - 1 \cdot 49$

$x = 7$

Step 4.2.1.1.3

Multiply $- 1$ by $49$ .

$y = 49 - 49$

$x = 7$

$y = 49 - 49$

$x = 7$

Step 4.2.1.2

Subtract $49$ from $49$ .

$y = 0$

$x = 7$

$y = 0$

$x = 7$

$y = 0$

$x = 7$

$y = 0$

$x = 7$

Step 5

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

$(7, 0)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(7, 0)$

Equation Form:

$x = 7, y = 0$

Step 7