Algebra Examples

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

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

Subtract $2 x$ from both sides of the equation.

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

Step 2.1.2

Subtract $2 x$ from $- 2 x$ .

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

Step 2.2

Add $3$ to both sides of the equation.

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

Step 2.3

Combine the opposite terms in $x^{2} - 4 x - 3 + 3$ .

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Step 2.3.1

Add $- 3$ and $3$ .

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

Step 2.3.2

Add $x^{2} - 4 x$ and $0$ .

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

Step 2.4

Factor $x$ out of $x^{2} - 4 x$ .

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Step 2.4.1

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

$x \cdot x - 4 x = 0$

Step 2.4.2

Factor $x$ out of $- 4 x$ .

$x \cdot x + x \cdot - 4 = 0$

Step 2.4.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$x (x - 4) = 0$

Step 2.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 4 = 0 + y = 2 x - 3$

Step 2.6

Set $x$ equal to $0$ .

$x = 0$

Step 2.7

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

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Step 2.7.1

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

$x - 4 = 0$

Step 2.7.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.8

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

$x = 0, 4$

Step 3

Evaluate $y$ when $x = 0$ .

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Step 3.1

Substitute $0$ for $x$ .

$y = 2 (0) - 3$

Step 3.2

Simplify $2 (0) - 3$ .

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Step 3.2.1

Multiply $2$ by $0$ .

$y = 0 - 3$

Step 3.2.2

Subtract $3$ from $0$ .

$y = - 3$

Step 4

Evaluate $y$ when $x = 4$ .

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Step 4.1

Substitute $4$ for $x$ .

$y = 2 (4) - 3$

Step 4.2

Simplify $2 (4) - 3$ .

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Step 4.2.1

Multiply $2$ by $4$ .

$y = 8 - 3$

Step 4.2.2

Subtract $3$ from $8$ .

$y = 5$

Step 5

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

$(0, - 3)$

$(4, 5)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(0, - 3), (4, 5)$

Equation Form:

$x = 0, y = - 3$

$x = 4, y = 5$

Step 7