Precalculus Examples

$a x + b y = 0$ , $a^{2} x + b^{2} y = 1$

Step 1

Solve the equation for $x$ .

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

$a x = - b y$

$a^{2} x + b^{2} y = 1$

Divide each term in $a x = - b y$ by $a$ and simplify.

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Divide each term in $a x = - b y$ by $a$ .

$\frac{a x}{a} = \frac{- b y}{a}$

$a^{2} x + b^{2} y = 1$

Simplify the left side.

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Cancel the common factor of $a$ .

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Cancel the common factor.

$\frac{a x}{a} = \frac{- b y}{a}$

$a^{2} x + b^{2} y = 1$

Divide $x$ by $1$ .

$x = \frac{- b y}{a}$

$a^{2} x + b^{2} y = 1$

$x = \frac{- b y}{a}$

$a^{2} x + b^{2} y = 1$

$x = \frac{- b y}{a}$

$a^{2} x + b^{2} y = 1$

Simplify the right side.

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Move the negative in front of the fraction.

$x = - \frac{b y}{a}$

$a^{2} x + b^{2} y = 1$

$x = - \frac{b y}{a}$

$a^{2} x + b^{2} y = 1$

$x = - \frac{b y}{a}$

$a^{2} x + b^{2} y = 1$

$x = - \frac{b y}{a}$

$a^{2} x + b^{2} y = 1$

Step 2

Solve the equation for $b$ .

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Simplify each term.

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Rewrite using the commutative property of multiplication.

$- a^{2} \frac{b y}{a} + b^{2} y = 1$

$x = - \frac{b y}{a}$

Cancel the common factor of $a$ .

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Factor $a$ out of $- a^{2}$ .

$a (- a) (\frac{b y}{a}) + b^{2} y = 1$

$x = - \frac{b y}{a}$

Cancel the common factor.

$a (- a) (\frac{b y}{a}) + b^{2} y = 1$

$x = - \frac{b y}{a}$

Rewrite the expression.

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

$x = - \frac{b y}{a}$

$- a b y + b^{2} y = 1$

$x = - \frac{b y}{a}$

$- a b y + b^{2} y = 1$

$x = - \frac{b y}{a}$

Subtract $1$ from both sides of the equation.

$- a b y + b^{2} y - 1 = 0$

$x = - \frac{b y}{a}$

Use the quadratic formula to find the solutions.

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

$x = - \frac{b y}{a}$

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

$\frac{a y \pm \sqrt{{(- a y)}^{2} - 4 \cdot (y \cdot - 1)}}{2 y}$

$x = - \frac{b y}{a}$

Simplify the numerator.

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Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- a y$ .

$b = \frac{a y \pm \sqrt{{(- a)}^{2} y^{2} - 4 \cdot y \cdot - 1}}{2 y}$

$x = - \frac{b y}{a}$

Apply the product rule to $- a$ .

$b = \frac{a y \pm \sqrt{{(- 1)}^{2} a^{2} y^{2} - 4 \cdot y \cdot - 1}}{2 y}$

$x = - \frac{b y}{a}$

$b = \frac{a y \pm \sqrt{{(- 1)}^{2} a^{2} y^{2} - 4 \cdot y \cdot - 1}}{2 y}$

$x = - \frac{b y}{a}$

Raise $- 1$ to the power of $2$ .

$b = \frac{a y \pm \sqrt{1 a^{2} y^{2} - 4 \cdot y \cdot - 1}}{2 y}$

$x = - \frac{b y}{a}$

Multiply $a^{2}$ by $1$ .

$b = \frac{a y \pm \sqrt{a^{2} y^{2} - 4 \cdot y \cdot - 1}}{2 y}$

$x = - \frac{b y}{a}$

Multiply $- 1$ by $- 4$ .

$b = \frac{a y \pm \sqrt{a^{2} y^{2} + 4 y}}{2 y}$

$x = - \frac{b y}{a}$

Factor $y$ out of $a^{2} y^{2} + 4 y$ .

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Factor $y$ out of $a^{2} y^{2}$ .

$b = \frac{a y \pm \sqrt{y (a^{2} y) + 4 y}}{2 y}$

$x = - \frac{b y}{a}$

Factor $y$ out of $4 y$ .

$b = \frac{a y \pm \sqrt{y (a^{2} y) + y \cdot 4}}{2 y}$

$x = - \frac{b y}{a}$

Factor $y$ out of $y (a^{2} y) + y \cdot 4$ .

$b = \frac{a y \pm \sqrt{y (a^{2} y + 4)}}{2 y}$

$x = - \frac{b y}{a}$

$b = \frac{a y \pm \sqrt{y (a^{2} y + 4)}}{2 y}$

$x = - \frac{b y}{a}$

$b = \frac{a y \pm \sqrt{y (a^{2} y + 4)}}{2 y}$

$x = - \frac{b y}{a}$

Change the $\pm$ to $+$ .

$b = \frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}$

$x = - \frac{b y}{a}$

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

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Simplify the numerator.

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Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- a y$ .

$b = \frac{a y \pm \sqrt{{(- a)}^{2} y^{2} - 4 \cdot y \cdot - 1}}{2 y}$

$x = - \frac{b y}{a}$

Apply the product rule to $- a$ .

$b = \frac{a y \pm \sqrt{{(- 1)}^{2} a^{2} y^{2} - 4 \cdot y \cdot - 1}}{2 y}$

$x = - \frac{b y}{a}$

$b = \frac{a y \pm \sqrt{{(- 1)}^{2} a^{2} y^{2} - 4 \cdot y \cdot - 1}}{2 y}$

$x = - \frac{b y}{a}$

Raise $- 1$ to the power of $2$ .

$b = \frac{a y \pm \sqrt{1 a^{2} y^{2} - 4 \cdot y \cdot - 1}}{2 y}$

$x = - \frac{b y}{a}$

Multiply $a^{2}$ by $1$ .

$b = \frac{a y \pm \sqrt{a^{2} y^{2} - 4 \cdot y \cdot - 1}}{2 y}$

$x = - \frac{b y}{a}$

Multiply $- 1$ by $- 4$ .

$b = \frac{a y \pm \sqrt{a^{2} y^{2} + 4 y}}{2 y}$

$x = - \frac{b y}{a}$

Factor $y$ out of $a^{2} y^{2} + 4 y$ .

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Factor $y$ out of $a^{2} y^{2}$ .

$b = \frac{a y \pm \sqrt{y (a^{2} y) + 4 y}}{2 y}$

$x = - \frac{b y}{a}$

Factor $y$ out of $4 y$ .

$b = \frac{a y \pm \sqrt{y (a^{2} y) + y \cdot 4}}{2 y}$

$x = - \frac{b y}{a}$

Factor $y$ out of $y (a^{2} y) + y \cdot 4$ .

$b = \frac{a y \pm \sqrt{y (a^{2} y + 4)}}{2 y}$

$x = - \frac{b y}{a}$

$b = \frac{a y \pm \sqrt{y (a^{2} y + 4)}}{2 y}$

$x = - \frac{b y}{a}$

$b = \frac{a y \pm \sqrt{y (a^{2} y + 4)}}{2 y}$

$x = - \frac{b y}{a}$

Change the $\pm$ to $-$ .

$b = \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}$

$x = - \frac{b y}{a}$

$b = \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}$

$x = - \frac{b y}{a}$

The final answer is the combination of both solutions.

$b = \frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}$

$b = \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}$

$x = - \frac{b y}{a}$

$b = \frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}$

$b = \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}$

$x = - \frac{b y}{a}$

Step 3

Simplify the right side.

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Multiply $2$ by $0$ .

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$b = \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{2 (0)}$

$a (- (\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) \cdot \frac{y}{a}) + (\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) \cdot y = 0$

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}, \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{2 (0)}$

Step 4

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

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$b = \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{2 (0)}$

Step 5

Simplify the left side.

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Simplify $a (- (\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) \cdot \frac{y}{a}) + (\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) \cdot y$ .

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Simplify each term.

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Rewrite using the commutative property of multiplication.

$- a (\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y (\frac{y}{a})}) + (\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) \cdot y = 0$

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$b = \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{2 (0)}$

Combine $\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}$ and $\frac{y}{a}$ .

$- a \frac{(\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y \cdot y})}{a} + (\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) \cdot y = 0$

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$b = \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{2 (0)}$

Cancel the common factor of $a$ .

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Factor $a$ out of $- a$ .

$a \cdot (- 1 \frac{(\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y \cdot y})}{a}) + (\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) \cdot y = 0$

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$b = \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{2 (0)}$

Cancel the common factor.

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$b = \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{2 (0)}$

Rewrite the expression.

$- (\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y \cdot y}) + (\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) \cdot y = 0$

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$b = \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{2 (0)}$

$- \frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y \cdot y} + (\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) y = 0$

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$b = \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{2 (0)}$

$- \frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y \cdot y} + (\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) y = 0$

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$b = \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{2 (0)}$

Reorder $- \frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y} y$ and $(\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) y$ .

$(\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) y - \frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y \cdot y} = 0$

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$b = \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{2 (0)}$

$(\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) y - \frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y \cdot y} = 0$

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$b = \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{2 (0)}$

$(\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) y - \frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y \cdot y} = 0$

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}, \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{2 (0)}$

Step 6

Simplify the right side.

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Multiply $2$ by $0$ .

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$b = \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$(\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) y - \frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y \cdot y} = 0$

$b = \frac{a (0) + \sqrt{(0) (a^{2} (0) + 4)}}{0}, \frac{a (0) - \sqrt{(0) (a^{2} (0) + 4)}}{0}$

$(\frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y}) y - \frac{a y + \sqrt{y (a^{2} y + 4)}}{2 y}, \frac{a y - \sqrt{y (a^{2} y + 4)}}{2 y \cdot y} = 0$