Precalculus Examples

$q = x y$ , $x + y = 50$

Step 1

Solve the equation for $x$ .

Tap for more steps...

Rewrite the equation as $x y = q$ .

$x y = q$

$x + y = 50$

Divide each term in $x y = q$ by $y$ and simplify.

Tap for more steps...

Divide each term in $x y = q$ by $y$ .

$\frac{x y}{y} = \frac{q}{y}$

$x + y = 50$

Simplify the left side.

Tap for more steps...

Cancel the common factor of $y$ .

Tap for more steps...

Cancel the common factor.

$\frac{x y}{y} = \frac{q}{y}$

$x + y = 50$

Divide $x$ by $1$ .

$x = \frac{q}{y}$

$x + y = 50$

$x = \frac{q}{y}$

$x + y = 50$

$x = \frac{q}{y}$

$x + y = 50$

$x = \frac{q}{y}$

$x + y = 50$

$x = \frac{q}{y}$

$x + y = 50$

Step 2

Solve the equation for $y$ .

Tap for more steps...

Remove parentheses.

$\frac{q}{y} + y = 50$

$x = \frac{q}{y}$

Find the LCD of the terms in the equation.

Tap for more steps...

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, 1, 1$

$x = \frac{q}{y}$

The LCM of one and any expression is the expression.

$x = \frac{q}{y}$

Multiply each term in $\frac{q}{y} + y = 50$ by $y$ to eliminate the fractions.

Tap for more steps...

Multiply each term in $\frac{q}{y} + y = 50$ by $y$ .

$\frac{q}{y} \cdot y + y \cdot y = 50 y$

$x = \frac{q}{y}$

Simplify the left side.

Tap for more steps...

Simplify each term.

Tap for more steps...

Cancel the common factor of $y$ .

Tap for more steps...

Cancel the common factor.

$\frac{q}{y} \cdot y + y \cdot y = 50 y$

$x = \frac{q}{y}$

Rewrite the expression.

$q + y \cdot y = 50 y$

$x = \frac{q}{y}$

$q + y \cdot y = 50 y$

$x = \frac{q}{y}$

Multiply $y$ by $y$ .

$q + y^{2} = 50 y$

$x = \frac{q}{y}$

$q + y^{2} = 50 y$

$x = \frac{q}{y}$

$q + y^{2} = 50 y$

$x = \frac{q}{y}$

$q + y^{2} = 50 y$

$x = \frac{q}{y}$

Solve the equation.

Tap for more steps...

Subtract $50 y$ from both sides of the equation.

$q + y^{2} - 50 y = 0$

$x = \frac{q}{y}$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

$x = \frac{q}{y}$

Substitute the values $a = 1$ , $b = - 50$ , and $c = q$ into the quadratic formula and solve for $y$ .

$\frac{50 \pm \sqrt{{(- 50)}^{2} - 4 \cdot (1 \cdot q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

Simplify.

Tap for more steps...

Simplify the numerator.

Tap for more steps...

Raise $- 50$ to the power of $2$ .

$y = \frac{50 \pm \sqrt{2500 - 4 \cdot 1 \cdot q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Multiply $- 4$ by $1$ .

$y = \frac{50 \pm \sqrt{2500 - 4 \cdot q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Factor $4$ out of $2500 - 4 q$ .

Tap for more steps...

Factor $4$ out of $2500$ .

$y = \frac{50 \pm \sqrt{4 (625) - 4 q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Factor $4$ out of $- 4 q$ .

$y = \frac{50 \pm \sqrt{4 (625) + 4 (- q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

Factor $4$ out of $4 (625) + 4 (- q)$ .

$y = \frac{50 \pm \sqrt{4 (625 - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

$y = \frac{50 \pm \sqrt{4 (625 - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

Rewrite $4 (625 - q)$ as $2^{2} (25^{2} - q)$ .

Tap for more steps...

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

$y = \frac{50 \pm \sqrt{2^{2} (625 - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

Rewrite $625$ as $25^{2}$ .

$y = \frac{50 \pm \sqrt{2^{2} (25^{2} - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

$y = \frac{50 \pm \sqrt{2^{2} (25^{2} - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

Pull terms out from under the radical.

$y = \frac{50 \pm 2 \sqrt{25^{2} - q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Raise $25$ to the power of $2$ .

$y = \frac{50 \pm 2 \sqrt{625 - q}}{2 \cdot 1}$

$x = \frac{q}{y}$

$y = \frac{50 \pm 2 \sqrt{625 - q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Multiply $2$ by $1$ .

$y = \frac{50 \pm 2 \sqrt{625 - q}}{2}$

$x = \frac{q}{y}$

Simplify $\frac{50 \pm 2 \sqrt{625 - q}}{2}$ .

$y = 25 \pm \sqrt{625 - q}$

$x = \frac{q}{y}$

$y = 25 \pm \sqrt{625 - q}$

$x = \frac{q}{y}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Simplify the numerator.

Tap for more steps...

Raise $- 50$ to the power of $2$ .

$y = \frac{50 \pm \sqrt{2500 - 4 \cdot 1 \cdot q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Multiply $- 4$ by $1$ .

$y = \frac{50 \pm \sqrt{2500 - 4 \cdot q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Factor $4$ out of $2500 - 4 q$ .

Tap for more steps...

Factor $4$ out of $2500$ .

$y = \frac{50 \pm \sqrt{4 (625) - 4 q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Factor $4$ out of $- 4 q$ .

$y = \frac{50 \pm \sqrt{4 (625) + 4 (- q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

Factor $4$ out of $4 (625) + 4 (- q)$ .

$y = \frac{50 \pm \sqrt{4 (625 - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

$y = \frac{50 \pm \sqrt{4 (625 - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

Rewrite $4 (625 - q)$ as $2^{2} (25^{2} - q)$ .

Tap for more steps...

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

$y = \frac{50 \pm \sqrt{2^{2} (625 - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

Rewrite $625$ as $25^{2}$ .

$y = \frac{50 \pm \sqrt{2^{2} (25^{2} - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

$y = \frac{50 \pm \sqrt{2^{2} (25^{2} - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

Pull terms out from under the radical.

$y = \frac{50 \pm 2 \sqrt{25^{2} - q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Raise $25$ to the power of $2$ .

$y = \frac{50 \pm 2 \sqrt{625 - q}}{2 \cdot 1}$

$x = \frac{q}{y}$

$y = \frac{50 \pm 2 \sqrt{625 - q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Multiply $2$ by $1$ .

$y = \frac{50 \pm 2 \sqrt{625 - q}}{2}$

$x = \frac{q}{y}$

Simplify $\frac{50 \pm 2 \sqrt{625 - q}}{2}$ .

$y = 25 \pm \sqrt{625 - q}$

$x = \frac{q}{y}$

Change the $\pm$ to $+$ .

$y = 25 + \sqrt{625 - q}$

$x = \frac{q}{y}$

$y = 25 + \sqrt{625 - q}$

$x = \frac{q}{y}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Simplify the numerator.

Tap for more steps...

Raise $- 50$ to the power of $2$ .

$y = \frac{50 \pm \sqrt{2500 - 4 \cdot 1 \cdot q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Multiply $- 4$ by $1$ .

$y = \frac{50 \pm \sqrt{2500 - 4 \cdot q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Factor $4$ out of $2500 - 4 q$ .

Tap for more steps...

Factor $4$ out of $2500$ .

$y = \frac{50 \pm \sqrt{4 (625) - 4 q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Factor $4$ out of $- 4 q$ .

$y = \frac{50 \pm \sqrt{4 (625) + 4 (- q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

Factor $4$ out of $4 (625) + 4 (- q)$ .

$y = \frac{50 \pm \sqrt{4 (625 - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

$y = \frac{50 \pm \sqrt{4 (625 - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

Rewrite $4 (625 - q)$ as $2^{2} (25^{2} - q)$ .

Tap for more steps...

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

$y = \frac{50 \pm \sqrt{2^{2} (625 - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

Rewrite $625$ as $25^{2}$ .

$y = \frac{50 \pm \sqrt{2^{2} (25^{2} - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

$y = \frac{50 \pm \sqrt{2^{2} (25^{2} - q)}}{2 \cdot 1}$

$x = \frac{q}{y}$

Pull terms out from under the radical.

$y = \frac{50 \pm 2 \sqrt{25^{2} - q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Raise $25$ to the power of $2$ .

$y = \frac{50 \pm 2 \sqrt{625 - q}}{2 \cdot 1}$

$x = \frac{q}{y}$

$y = \frac{50 \pm 2 \sqrt{625 - q}}{2 \cdot 1}$

$x = \frac{q}{y}$

Multiply $2$ by $1$ .

$y = \frac{50 \pm 2 \sqrt{625 - q}}{2}$

$x = \frac{q}{y}$

Simplify $\frac{50 \pm 2 \sqrt{625 - q}}{2}$ .

$y = 25 \pm \sqrt{625 - q}$

$x = \frac{q}{y}$

Change the $\pm$ to $-$ .

$y = 25 - \sqrt{625 - q}$

$x = \frac{q}{y}$

$y = 25 - \sqrt{625 - q}$

$x = \frac{q}{y}$

The final answer is the combination of both solutions.

$y = 25 + \sqrt{625 - q}$

$y = 25 - \sqrt{625 - q}$

$x = \frac{q}{y}$

$y = 25 + \sqrt{625 - q}$

$y = 25 - \sqrt{625 - q}$

$x = \frac{q}{y}$

$y = 25 + \sqrt{625 - q}$

$y = 25 - \sqrt{625 - q}$

$x = \frac{q}{y}$

Step 3

The simplified system is the arbitrary solution of the original system of equations.

$x = \frac{q}{(25 + \sqrt{625 - q}, 25 - \sqrt{625 - q})}$

$y = 25 + \sqrt{625 - q}$

$y = 25 - \sqrt{625 - q}$