Algebra Examples

$y = - x^{2} + 6 x$ $4 y = 21 - x$

Step 1

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

Tap for more steps...

Step 1.1

Replace all occurrences of $y$ in $4 y = 21 - x$ with $- x^{2} + 6 x$ .

$4 (- x^{2} + 6 x) = 21 - x$

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

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Simplify $4 (- x^{2} + 6 x)$ .

Tap for more steps...

Step 1.2.1.1

Apply the distributive property.

$4 (- x^{2}) + 4 (6 x) = 21 - x$

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

Step 1.2.1.2

Multiply.

Tap for more steps...

Step 1.2.1.2.1

Multiply $- 1$ by $4$ .

$- 4 x^{2} + 4 (6 x) = 21 - x$

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

Step 1.2.1.2.2

Multiply $6$ by $4$ .

$- 4 x^{2} + 24 x = 21 - x$

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

$- 4 x^{2} + 24 x = 21 - x$

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

$- 4 x^{2} + 24 x = 21 - x$

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

$- 4 x^{2} + 24 x = 21 - x$

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

$- 4 x^{2} + 24 x = 21 - x$

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

Step 2

Solve for $x$ in $- 4 x^{2} + 24 x = 21 - x$ .

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

Add $x$ to both sides of the equation.

$- 4 x^{2} + 24 x + x = 21$

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

Step 2.1.2

Add $24 x$ and $x$ .

$- 4 x^{2} + 25 x = 21$

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

$- 4 x^{2} + 25 x = 21$

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

Step 2.2

Subtract $21$ from both sides of the equation.

$- 4 x^{2} + 25 x - 21 = 0$

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

Step 2.3

Factor the left side of the equation.

Tap for more steps...

Step 2.3.1

Factor $- 1$ out of $- 4 x^{2} + 25 x - 21$ .

Tap for more steps...

Step 2.3.1.1

Factor $- 1$ out of $- 4 x^{2}$ .

$- (4 x^{2}) + 25 x - 21 = 0$

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

Step 2.3.1.2

Factor $- 1$ out of $25 x$ .

$- (4 x^{2}) - (- 25 x) - 21 = 0$

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

Step 2.3.1.3

Rewrite $- 21$ as $- 1 (21)$ .

$- (4 x^{2}) - (- 25 x) - 1 \cdot 21 = 0$

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

Step 2.3.1.4

Factor $- 1$ out of $- (4 x^{2}) - (- 25 x)$ .

$- (4 x^{2} - 25 x) - 1 \cdot 21 = 0$

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

Step 2.3.1.5

Factor $- 1$ out of $- (4 x^{2} - 25 x) - 1 (21)$ .

$- (4 x^{2} - 25 x + 21) = 0$

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

$- (4 x^{2} - 25 x + 21) = 0$

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

Step 2.3.2

Factor.

Tap for more steps...

Step 2.3.2.1

Factor by grouping.

Tap for more steps...

Step 2.3.2.1.1

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 $a \cdot c = 4 \cdot 21 = 84$ and whose sum is $b = - 25$ .

Tap for more steps...

Step 2.3.2.1.1.1

Factor $- 25$ out of $- 25 x$ .

$- (4 x^{2} - 25 x + 21) = 0$

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

Step 2.3.2.1.1.2

Rewrite $- 25$ as $- 4$ plus $- 21$

$- (4 x^{2} + (- 4 - 21) x + 21) = 0$

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

Step 2.3.2.1.1.3

Apply the distributive property.

$- (4 x^{2} - 4 x - 21 x + 21) = 0$

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

$- (4 x^{2} - 4 x - 21 x + 21) = 0$

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

Step 2.3.2.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.3.2.1.2.1

Group the first two terms and the last two terms.

$- ((4 x^{2} - 4 x) - 21 x + 21) = 0$

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

Step 2.3.2.1.2.2

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

$- (4 x (x - 1) - 21 (x - 1)) = 0$

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

$- (4 x (x - 1) - 21 (x - 1)) = 0$

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

Step 2.3.2.1.3

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

$- ((x - 1) (4 x - 21)) = 0$

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

$- ((x - 1) (4 x - 21)) = 0$

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

Step 2.3.2.2

Remove unnecessary parentheses.

$- (x - 1) (4 x - 21) = 0$

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

$- (x - 1) (4 x - 21) = 0$

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

$- (x - 1) (4 x - 21) = 0$

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

Step 2.4

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

$x - 1 = 0$

$4 x - 21 = 0$

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

Step 2.5

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

Tap for more steps...

Step 2.5.1

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

$x - 1 = 0$

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

Step 2.5.2

Add $1$ to both sides of the equation.

$x = 1$

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

$x = 1$

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

Step 2.6

Set $4 x - 21$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.6.1

Set $4 x - 21$ equal to $0$ .

$4 x - 21 = 0$

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

Step 2.6.2

Solve $4 x - 21 = 0$ for $x$ .

Tap for more steps...

Step 2.6.2.1

Add $21$ to both sides of the equation.

$4 x = 21$

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

Step 2.6.2.2

Divide each term in $4 x = 21$ by $4$ and simplify.

Tap for more steps...

Step 2.6.2.2.1

Divide each term in $4 x = 21$ by $4$ .

$\frac{4 x}{4} = \frac{21}{4}$

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

Step 2.6.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.6.2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.6.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{21}{4}$

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

Step 2.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{21}{4}$

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

$x = \frac{21}{4}$

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

$x = \frac{21}{4}$

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

$x = \frac{21}{4}$

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

$x = \frac{21}{4}$

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

$x = \frac{21}{4}$

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

Step 2.7

The final solution is all the values that make $- (x - 1) (4 x - 21) = 0$ true.

$x = 1, \frac{21}{4}$

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

$x = 1, \frac{21}{4}$

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

Step 3

Replace all occurrences of $x$ with $1$ in each equation.

Tap for more steps...

Step 3.1

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

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

$x = 1$

Step 3.2

Simplify the right side.

Tap for more steps...

Step 3.2.1

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

Tap for more steps...

Step 3.2.1.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1.1

One to any power is one.

$y = - 1 \cdot 1 + 6 (1)$

$x = 1$

Step 3.2.1.1.2

Multiply $- 1$ by $1$ .

$y = - 1 + 6 (1)$

$x = 1$

Step 3.2.1.1.3

Multiply $6$ by $1$ .

$y = - 1 + 6$

$x = 1$

$y = - 1 + 6$

$x = 1$

Step 3.2.1.2

Add $- 1$ and $6$ .

$y = 5$

$x = 1$

$y = 5$

$x = 1$

$y = 5$

$x = 1$

$y = 5$

$x = 1$

Step 4

Replace all occurrences of $x$ with $\frac{21}{4}$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $x$ in $y = - x^{2} + 6 x$ with $\frac{21}{4}$ .

$y = - {(\frac{21}{4})}^{2} + 6 (\frac{21}{4})$

$x = \frac{21}{4}$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $- {(\frac{21}{4})}^{2} + 6 (\frac{21}{4})$ .

Tap for more steps...

Step 4.2.1.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1.1

Apply the product rule to $\frac{21}{4}$ .

$y = - \frac{21^{2}}{4^{2}} + 6 (\frac{21}{4})$

$x = \frac{21}{4}$

Step 4.2.1.1.2

Raise $21$ to the power of $2$ .

$y = - \frac{441}{4^{2}} + 6 (\frac{21}{4})$

$x = \frac{21}{4}$

Step 4.2.1.1.3

Raise $4$ to the power of $2$ .

$y = - \frac{441}{16} + 6 (\frac{21}{4})$

$x = \frac{21}{4}$

Step 4.2.1.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.1.1.4.1

Factor $2$ out of $6$ .

$y = - \frac{441}{16} + 2 (3) (\frac{21}{4})$

$x = \frac{21}{4}$

Step 4.2.1.1.4.2

Factor $2$ out of $4$ .

$y = - \frac{441}{16} + 2 \cdot (3 (\frac{21}{2 \cdot 2}))$

$x = \frac{21}{4}$

Step 4.2.1.1.4.3

Cancel the common factor.

$y = - \frac{441}{16} + 2 \cdot (3 (\frac{21}{2 \cdot 2}))$

$x = \frac{21}{4}$

Step 4.2.1.1.4.4

Rewrite the expression.

$y = - \frac{441}{16} + 3 (\frac{21}{2})$

$x = \frac{21}{4}$

$y = - \frac{441}{16} + 3 (\frac{21}{2})$

$x = \frac{21}{4}$

Step 4.2.1.1.5

Combine $3$ and $\frac{21}{2}$ .

$y = - \frac{441}{16} + \frac{3 \cdot 21}{2}$

$x = \frac{21}{4}$

Step 4.2.1.1.6

Multiply $3$ by $21$ .

$y = - \frac{441}{16} + \frac{63}{2}$

$x = \frac{21}{4}$

$y = - \frac{441}{16} + \frac{63}{2}$

$x = \frac{21}{4}$

Step 4.2.1.2

To write $\frac{63}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = - \frac{441}{16} + \frac{63}{2} \cdot \frac{8}{8}$

$x = \frac{21}{4}$

Step 4.2.1.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.2.1.3.1

Multiply $\frac{63}{2}$ by $\frac{8}{8}$ .

$y = - \frac{441}{16} + \frac{63 \cdot 8}{2 \cdot 8}$

$x = \frac{21}{4}$

Step 4.2.1.3.2

Multiply $2$ by $8$ .

$y = - \frac{441}{16} + \frac{63 \cdot 8}{16}$

$x = \frac{21}{4}$

$y = - \frac{441}{16} + \frac{63 \cdot 8}{16}$

$x = \frac{21}{4}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$y = \frac{- 441 + 63 \cdot 8}{16}$

$x = \frac{21}{4}$

Step 4.2.1.5

Simplify the numerator.

Tap for more steps...

Step 4.2.1.5.1

Multiply $63$ by $8$ .

$y = \frac{- 441 + 504}{16}$

$x = \frac{21}{4}$

Step 4.2.1.5.2

Add $- 441$ and $504$ .

$y = \frac{63}{16}$

$x = \frac{21}{4}$

$y = \frac{63}{16}$

$x = \frac{21}{4}$

$y = \frac{63}{16}$

$x = \frac{21}{4}$

$y = \frac{63}{16}$

$x = \frac{21}{4}$

$y = \frac{63}{16}$

$x = \frac{21}{4}$

Step 5

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

$(1, 5)$

$(\frac{21}{4}, \frac{63}{16})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(1, 5), (\frac{21}{4}, \frac{63}{16})$

Equation Form:

$x = 1, y = 5$

$x = \frac{21}{4}, y = \frac{63}{16}$

Step 7