Algebra Examples

$y = x^{2} - 3 x - 4$ , $x = y + 8$

Step 1

Replace all occurrences of $y$ with $x^{2} - 3 x - 4$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $x = y + 8$ with $x^{2} - 3 x - 4$ .

$x = (x^{2} - 3 x - 4) + 8$

$y = x^{2} - 3 x - 4$

Step 1.2

Simplify the right side.

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Step 1.2.1

Add $- 4$ and $8$ .

$x = x^{2} - 3 x + 4$

$y = x^{2} - 3 x - 4$

$x = x^{2} - 3 x + 4$

$y = x^{2} - 3 x - 4$

$x = x^{2} - 3 x + 4$

$y = x^{2} - 3 x - 4$

Step 2

Solve for $x$ in $x = x^{2} - 3 x + 4$ .

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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.

$x^{2} - 3 x + 4 = x$

$y = x^{2} - 3 x - 4$

Step 2.2

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

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Step 2.2.1

Subtract $x$ from both sides of the equation.

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

$y = x^{2} - 3 x - 4$

Step 2.2.2

Subtract $x$ from $- 3 x$ .

$x^{2} - 4 x + 4 = 0$

$y = x^{2} - 3 x - 4$

$x^{2} - 4 x + 4 = 0$

$y = x^{2} - 3 x - 4$

Step 2.3

Factor using the perfect square rule.

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Step 2.3.1

Rewrite $4$ as $2^{2}$ .

$x^{2} - 4 x + 2^{2} = 0$

$y = x^{2} - 3 x - 4$

Step 2.3.2

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

$4 x = 2 \cdot x \cdot 2$

$y = x^{2} - 3 x - 4$

Step 2.3.3

Rewrite the polynomial.

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

$y = x^{2} - 3 x - 4$

Step 2.3.4

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

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

$y = x^{2} - 3 x - 4$

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

$y = x^{2} - 3 x - 4$

Step 2.4

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

$x - 2 = 0$

$y = x^{2} - 3 x - 4$

Step 2.5

Add $2$ to both sides of the equation.

$x = 2$

$y = x^{2} - 3 x - 4$

$x = 2$

$y = x^{2} - 3 x - 4$

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 $y = x^{2} - 3 x - 4$ with $2$ .

$y = {(2)}^{2} - 3 \cdot 2 - 4$

$x = 2$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify ${(2)}^{2} - 3 \cdot 2 - 4$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

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

$y = 4 - 3 \cdot 2 - 4$

$x = 2$

Step 3.2.1.1.2

Multiply $- 3$ by $2$ .

$y = 4 - 6 - 4$

$x = 2$

$y = 4 - 6 - 4$

$x = 2$

Step 3.2.1.2

Simplify by subtracting numbers.

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Step 3.2.1.2.1

Subtract $6$ from $4$ .

$y = - 2 - 4$

$x = 2$

Step 3.2.1.2.2

Subtract $4$ from $- 2$ .

$y = - 6$

$x = 2$

$y = - 6$

$x = 2$

$y = - 6$

$x = 2$

$y = - 6$

$x = 2$

$y = - 6$

$x = 2$

Step 4

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

$(2, - 6)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(2, - 6)$

Equation Form:

$x = 2, y = - 6$

Step 6