Algebra Examples

$y = 3 x^{2} - 6 x + 1$ $y - x + 1 = 0$

Step 1

Replace all occurrences of $y$ with $3 x^{2} - 6 x + 1$ in each equation.

Tap for more steps...

Step 1.1

Replace all occurrences of $y$ in $y - x + 1 = 0$ with $3 x^{2} - 6 x + 1$ .

$(3 x^{2} - 6 x + 1) - x + 1 = 0$

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

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Simplify $(3 x^{2} - 6 x + 1) - x + 1$ .

Tap for more steps...

Step 1.2.1.1

Subtract $x$ from $- 6 x$ .

$3 x^{2} - 7 x + 1 + 1 = 0$

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

Step 1.2.1.2

Add $1$ and $1$ .

$3 x^{2} - 7 x + 2 = 0$

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

$3 x^{2} - 7 x + 2 = 0$

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

$3 x^{2} - 7 x + 2 = 0$

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

$3 x^{2} - 7 x + 2 = 0$

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

Step 2

Solve for $x$ in $3 x^{2} - 7 x + 2 = 0$ .

Tap for more steps...

Step 2.1

Factor by grouping.

Tap for more steps...

Step 2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 2 = 6$ and whose sum is $b = - 7$ .

Tap for more steps...

Step 2.1.1.1

Factor $- 7$ out of $- 7 x$ .

$3 x^{2} - 7 x + 2 = 0$

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

Step 2.1.1.2

Rewrite $- 7$ as $- 1$ plus $- 6$

$3 x^{2} + (- 1 - 6) x + 2 = 0$

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

Step 2.1.1.3

Apply the distributive property.

$3 x^{2} - 1 x - 6 x + 2 = 0$

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

$3 x^{2} - 1 x - 6 x + 2 = 0$

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

Step 2.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.1.2.1

Group the first two terms and the last two terms.

$(3 x^{2} - 1 x) - 6 x + 2 = 0$

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

Step 2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$x (3 x - 1) - 2 (3 x - 1) = 0$

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

$x (3 x - 1) - 2 (3 x - 1) = 0$

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

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(3 x - 1) (x - 2) = 0$

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

$(3 x - 1) (x - 2) = 0$

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

Step 2.2

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

$3 x - 1 = 0$

$x - 2 = 0$

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

Step 2.3

Set $3 x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.3.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

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

Step 2.3.2

Solve $3 x - 1 = 0$ for $x$ .

Tap for more steps...

Step 2.3.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

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

Step 2.3.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

Tap for more steps...

Step 2.3.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

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

Step 2.3.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

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

Step 2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

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

Step 2.4

Set $x - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.4.1

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

$x - 2 = 0$

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

Step 2.4.2

Add $2$ to both sides of the equation.

$x = 2$

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

$x = 2$

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

Step 2.5

The final solution is all the values that make $(3 x - 1) (x - 2) = 0$ true.

$x = \frac{1}{3}, 2$

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

$x = \frac{1}{3}, 2$

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

Step 3

Replace all occurrences of $x$ with $\frac{1}{3}$ in each equation.

Tap for more steps...

Step 3.1

Replace all occurrences of $x$ in $y = 3 x^{2} - 6 x + 1$ with $\frac{1}{3}$ .

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

$x = \frac{1}{3}$

Step 3.2

Simplify the right side.

Tap for more steps...

Step 3.2.1

Simplify $3 {(\frac{1}{3})}^{2} - 6 (\frac{1}{3}) + 1$ .

Tap for more steps...

Step 3.2.1.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1.1

Apply the product rule to $\frac{1}{3}$ .

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

$x = \frac{1}{3}$

Step 3.2.1.1.2

One to any power is one.

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

$x = \frac{1}{3}$

Step 3.2.1.1.3

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

$y = 3 (\frac{1}{9}) - 6 (\frac{1}{3}) + 1$

$x = \frac{1}{3}$

Step 3.2.1.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.1.4.1

Factor $3$ out of $9$ .

$y = 3 (\frac{1}{3 (3)}) - 6 (\frac{1}{3}) + 1$

$x = \frac{1}{3}$

Step 3.2.1.1.4.2

Cancel the common factor.

$y = 3 (\frac{1}{3 \cdot 3}) - 6 (\frac{1}{3}) + 1$

$x = \frac{1}{3}$

Step 3.2.1.1.4.3

Rewrite the expression.

$y = \frac{1}{3} - 6 (\frac{1}{3}) + 1$

$x = \frac{1}{3}$

$y = \frac{1}{3} - 6 (\frac{1}{3}) + 1$

$x = \frac{1}{3}$

Step 3.2.1.1.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.1.5.1

Factor $3$ out of $- 6$ .

$y = \frac{1}{3} + 3 (- 2) (\frac{1}{3}) + 1$

$x = \frac{1}{3}$

Step 3.2.1.1.5.2

Cancel the common factor.

$y = \frac{1}{3} + 3 \cdot (- 2 (\frac{1}{3})) + 1$

$x = \frac{1}{3}$

Step 3.2.1.1.5.3

Rewrite the expression.

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

Step 3.2.1.2

Find the common denominator.

Tap for more steps...

Step 3.2.1.2.1

Write $- 2$ as a fraction with denominator $1$ .

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

$x = \frac{1}{3}$

Step 3.2.1.2.2

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

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

$x = \frac{1}{3}$

Step 3.2.1.2.3

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

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

$x = \frac{1}{3}$

Step 3.2.1.2.4

Write $1$ as a fraction with denominator $1$ .

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

$x = \frac{1}{3}$

Step 3.2.1.2.5

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

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

$x = \frac{1}{3}$

Step 3.2.1.2.6

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

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

Step 3.2.1.3

Combine the numerators over the common denominator.

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

$x = \frac{1}{3}$

Step 3.2.1.4

Simplify the expression.

Tap for more steps...

Step 3.2.1.4.1

Multiply $- 2$ by $3$ .

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

$x = \frac{1}{3}$

Step 3.2.1.4.2

Subtract $6$ from $1$ .

$y = \frac{- 5 + 3}{3}$

$x = \frac{1}{3}$

Step 3.2.1.4.3

Add $- 5$ and $3$ .

$y = \frac{- 2}{3}$

$x = \frac{1}{3}$

Step 3.2.1.4.4

Move the negative in front of the fraction.

$y = - \frac{2}{3}$

$x = \frac{1}{3}$

$y = - \frac{2}{3}$

$x = \frac{1}{3}$

$y = - \frac{2}{3}$

$x = \frac{1}{3}$

$y = - \frac{2}{3}$

$x = \frac{1}{3}$

$y = - \frac{2}{3}$

$x = \frac{1}{3}$

Step 4

Replace all occurrences of $x$ with $2$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $x$ in $y = 3 x^{2} - 6 x + 1$ with $2$ .

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

$x = 2$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $3 {(2)}^{2} - 6 \cdot 2 + 1$ .

Tap for more steps...

Step 4.2.1.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1.1

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

$y = 3 \cdot 4 - 6 \cdot 2 + 1$

$x = 2$

Step 4.2.1.1.2

Multiply $3$ by $4$ .

$y = 12 - 6 \cdot 2 + 1$

$x = 2$

Step 4.2.1.1.3

Multiply $- 6$ by $2$ .

$y = 12 - 12 + 1$

$x = 2$

$y = 12 - 12 + 1$

$x = 2$

Step 4.2.1.2

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.1.2.1

Subtract $12$ from $12$ .

$y = 0 + 1$

$x = 2$

Step 4.2.1.2.2

Add $0$ and $1$ .

$y = 1$

$x = 2$

$y = 1$

$x = 2$

$y = 1$

$x = 2$

$y = 1$

$x = 2$

$y = 1$

$x = 2$

Step 5

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

$(\frac{1}{3}, - \frac{2}{3})$

$(2, 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{1}{3}, - \frac{2}{3}), (2, 1)$

Equation Form:

$x = \frac{1}{3}, y = - \frac{2}{3}$

$x = 2, y = 1$

Step 7