Precalculus Examples

$a x + b y = 1$ , $b x + a y = 1$

Step 1

Solve the equation for $x$ .

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

$a x = 1 - b y$

$b x + a y = 1$

Divide each term in $a x = 1 - b y$ by $a$ and simplify.

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Divide each term in $a x = 1 - b y$ by $a$ .

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

$b x + a y = 1$

Simplify the left side.

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Cancel the common factor of $a$ .

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Cancel the common factor.

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

$b x + a y = 1$

Divide $x$ by $1$ .

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

$b x + a y = 1$

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

$b x + a y = 1$

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

$b x + a y = 1$

Simplify the right side.

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Move the negative in front of the fraction.

$x = \frac{1}{a} - \frac{b y}{a}$

$b x + a y = 1$

$x = \frac{1}{a} - \frac{b y}{a}$

$b x + a y = 1$

$x = \frac{1}{a} - \frac{b y}{a}$

$b x + a y = 1$

$x = \frac{1}{a} - \frac{b y}{a}$

$b x + a y = 1$

Step 2

Solve the equation for $b$ .

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Simplify each term.

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Apply the distributive property.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Combine $b$ and $\frac{1}{a}$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite using the commutative property of multiplication.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $- b \frac{b y}{a}$ .

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Combine $\frac{b y}{a}$ and $b$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Raise $b$ to the power of $1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Raise $b$ to the power of $1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Add $1$ and $1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Find the LCD of the terms in the equation.

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

$a, a, 1, 1$

$x = \frac{1}{a} - \frac{b y}{a}$

Since $a, a, 1, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1, 1$ then find LCM for the variable part $a^{1}, a^{1}$ .

Since $a, a, 1, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1, 1$ then find LCM for the variable part a,a.

$x = \frac{1}{a} - \frac{b y}{a}$

The LCM is the smallest positive number that all of the numbers divide into evenly.

$1$ List the prime factors of each number.

$2$ Multiply each factor the greatest number of times it occurs in either number.

$x = \frac{1}{a} - \frac{b y}{a}$

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

$x = \frac{1}{a} - \frac{b y}{a}$

The factor for $a^{1}$ is $a$ itself.

$a = a$

a occurs $1$ time.

$x = \frac{1}{a} - \frac{b y}{a}$

The LCM of $a^{1}, a^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply each term in $\frac{b}{a} - \frac{b^{2} y}{a} + a y = 1$ by $a$ to eliminate the fractions.

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Multiply each term in $\frac{b}{a} - \frac{b^{2} y}{a} + a y = 1$ by $a$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Simplify the left side.

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Simplify each term.

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Cancel the common factor of $a$ .

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Cancel the common factor.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite the expression.

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Cancel the common factor of $a$ .

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Move the leading negative in $- \frac{b^{2} y}{a}$ into the numerator.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Cancel the common factor.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite the expression.

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $a$ by $a$ by adding the exponents.

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Move $a$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $a$ by $a$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Simplify the right side.

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Multiply $a$ by $1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Solve the equation.

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Subtract $a$ from both sides of the equation.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Use the quadratic formula to find the solutions.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Substitute the values $a = - y$ , $b = 1$ , and $c = a^{2} y - a$ into the quadratic formula and solve for $b$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Simplify.

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Simplify the numerator.

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One to any power is one.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $- 4$ by $- 1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Apply the distributive property.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $y$ by $y$ by adding the exponents.

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Move $y$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $y$ by $y$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite using the commutative property of multiplication.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $4$ by $- 1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Factor using the perfect square rule.

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Rearrange terms.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite $1$ as $1^{2}$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite $4 y^{2} a^{2}$ as ${(2 y a)}^{2}$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 y a = 2 \cdot 1 \cdot (2 y a)$

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite the polynomial.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 1$ and $b = 2 y a$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Pull terms out from under the radical, assuming positive real numbers.

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $- 1$ by $2$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Simplify $\frac{- 1 \pm (1 - 2 y a)}{- 2 y}$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Simplify the numerator.

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Apply the distributive property.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $- 1$ by $- 1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $2$ by $- 1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

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Simplify the numerator.

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One to any power is one.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $- 4$ by $- 1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Apply the distributive property.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $y$ by $y$ by adding the exponents.

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Move $y$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $y$ by $y$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite using the commutative property of multiplication.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $4$ by $- 1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Factor using the perfect square rule.

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Rearrange terms.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite $1$ as $1^{2}$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite $4 y^{2} a^{2}$ as ${(2 y a)}^{2}$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 y a = 2 \cdot 1 \cdot (2 y a)$

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite the polynomial.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 1$ and $b = 2 y a$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Pull terms out from under the radical, assuming positive real numbers.

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $- 1$ by $2$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Simplify $\frac{- 1 \pm (1 - 2 y a)}{- 2 y}$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Simplify the numerator.

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Apply the distributive property.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $- 1$ by $- 1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $2$ by $- 1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Change the $\pm$ to $+$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Simplify the numerator.

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Add $1$ and $1$ .

$b = \frac{2 - 2 y a}{2 y}$

$x = \frac{1}{a} - \frac{b y}{a}$

Factor $2$ out of $2 - 2 y a$ .

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Factor $2$ out of $2$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Factor $2$ out of $- 2 y a$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Factor $2$ out of $2 (1) + 2 (- y a)$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Cancel the common factor of $2$ .

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Cancel the common factor.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite the expression.

$b = \frac{1 - y a}{y}$

$x = \frac{1}{a} - \frac{b y}{a}$

$b = \frac{1 - y a}{y}$

$x = \frac{1}{a} - \frac{b y}{a}$

$b = \frac{1 - y a}{y}$

$x = \frac{1}{a} - \frac{b y}{a}$

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

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Simplify the numerator.

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One to any power is one.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $- 4$ by $- 1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Apply the distributive property.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $y$ by $y$ by adding the exponents.

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Move $y$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $y$ by $y$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite using the commutative property of multiplication.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $4$ by $- 1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Factor using the perfect square rule.

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Rearrange terms.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite $1$ as $1^{2}$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite $4 y^{2} a^{2}$ as ${(2 y a)}^{2}$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 y a = 2 \cdot 1 \cdot (2 y a)$

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite the polynomial.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 1$ and $b = 2 y a$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Pull terms out from under the radical, assuming positive real numbers.

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $- 1$ by $2$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Simplify $\frac{- 1 \pm (1 - 2 y a)}{- 2 y}$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Simplify the numerator.

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Apply the distributive property.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $- 1$ by $- 1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $2$ by $- 1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Change the $\pm$ to $-$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Simplify the numerator.

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Apply the distributive property.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $- 1$ by $1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Multiply $- 2$ by $- 1$ .

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

$x = \frac{1}{a} - \frac{b y}{a}$

Subtract $1$ from $1$ .

$b = \frac{0 + 2 y a}{2 y}$

$x = \frac{1}{a} - \frac{b y}{a}$

Add $0$ and $2 y a$ .

$b = \frac{2 y a}{2 y}$

$x = \frac{1}{a} - \frac{b y}{a}$

$b = \frac{2 y a}{2 y}$

$x = \frac{1}{a} - \frac{b y}{a}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$b = \frac{2 y a}{2 y}$

$x = \frac{1}{a} - \frac{b y}{a}$

Rewrite the expression.

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

$x = \frac{1}{a} - \frac{b y}{a}$

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

$x = \frac{1}{a} - \frac{b y}{a}$

Cancel the common factor of $y$ .

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Cancel the common factor.

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

$x = \frac{1}{a} - \frac{b y}{a}$

Divide $a$ by $1$ .

$b = a$

$x = \frac{1}{a} - \frac{b y}{a}$

$b = a$

$x = \frac{1}{a} - \frac{b y}{a}$

$b = a$

$x = \frac{1}{a} - \frac{b y}{a}$

The final answer is the combination of both solutions.

$b = \frac{1 - y a}{y}$

$b = a$

$x = \frac{1}{a} - \frac{b y}{a}$

$b = \frac{1 - y a}{y}$

$b = a$

$x = \frac{1}{a} - \frac{b y}{a}$

$b = \frac{1 - y a}{y}$

$b = a$

$x = \frac{1}{a} - \frac{b y}{a}$

Step 3

Simplify $a (\frac{1}{a} - \frac{1 - y a}{y}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y$ .

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Simplify each term.

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To write $\frac{1}{a}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$a (\frac{1}{a} \cdot \frac{y}{y} - \frac{1 - y a}{y}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

To write $- \frac{1 - y a}{y}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$a (\frac{1}{a} \cdot \frac{y}{y} - \frac{1 - y a}{y} \cdot \frac{a}{a}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

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

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Multiply $\frac{1}{a}$ by $\frac{y}{y}$ .

$a (\frac{y}{a y} - \frac{1 - y a}{y} \cdot \frac{a}{a}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

Multiply $\frac{1 - y a}{y}$ by $\frac{a}{a}$ .

$a (\frac{y}{a y} - \frac{(1 - y a) a}{y a}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

Reorder the factors of $y a$ .

$a (\frac{y}{a y} - \frac{(1 - y a) a}{a y}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

$a (\frac{y}{a y} - \frac{(1 - y a) a}{a y}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

Combine the numerators over the common denominator.

$a (\frac{y - (1 - y a) a}{a y}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

Simplify the numerator.

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Apply the distributive property.

$a (\frac{y + (- 1 \cdot 1 - (- y a)) a}{a y}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

Multiply $- 1$ by $1$ .

$a (\frac{y + (- 1 - (- y a)) a}{a y}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

Multiply $- (- y a)$ .

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Multiply $- 1$ by $- 1$ .

$a (\frac{y + (- 1 + 1 (y a)) a}{a y}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

Multiply $y$ by $1$ .

$a (\frac{y + (- 1 + y a) a}{a y}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

$a (\frac{y + (- 1 + y a) a}{a y}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

Apply the distributive property.

$a (\frac{y - 1 a + y a \cdot a}{a y}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

Rewrite $- 1 a$ as $- a$ .

$a (\frac{y - a + y a \cdot a}{a y}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

Multiply $a$ by $a$ by adding the exponents.

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Move $a$ .

$a (\frac{y - a + y (a \cdot a)}{a y}, a \cdot \frac{y}{a}) + (\frac{1 - y a}{y}, a) \cdot y = 1$

$b = \frac{1 - y a}{y}$

$b = a$

Multiply $a$ by $a$ .

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

$b = \frac{1 - y a}{y}$

$b = a$

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

$b = \frac{1 - y a}{y}$

$b = a$

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

$b = \frac{1 - y a}{y}$

$b = a$

Cancel the common factor of $a$ .

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Cancel the common factor.

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

$b = \frac{1 - y a}{y}$

$b = a$

Rewrite the expression.

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

$b = \frac{1 - y a}{y}$

$b = a$

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

$b = \frac{1 - y a}{y}$

$b = a$

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

$b = \frac{1 - y a}{y}$

$b = a$

Reorder factors in $a (\frac{y - a + y a^{2}}{a y}, y) + (\frac{1 - y a}{y}, a) y$ .

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

$b = \frac{1 - y a}{y}$

$b = a$

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

$b = \frac{1 - y a}{y}$

$b = a$

Step 4

Simplify the right side.

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Simplify $\frac{1 - 1 \cdot a}{1}, a$ .

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Divide $1 - 1 \cdot a$ by $1$ .

$b = 1 - 1 \cdot a$

$b = a$

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

Rewrite $- 1 a$ as $- a$ .

$b = 1 - a$

$b = a$

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

$b = 1 - a$

$b = a$

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

$b = 1 - a, a$

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

Step 5

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

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

$b = 1 - a$

$b = a$

Step 6

Simplify the left side.

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Simplify $a (\frac{y - a + y a^{2}}{a y}, y) + y (\frac{1 - y a}{y}, a)$ .

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Reorder $y$ and $a^{2}$ .

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

$b = 1 - a$

$b = a$

Move $- a$ .

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

$b = 1 - a$

$b = a$

Reorder $y$ and $a^{2} y$ .

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

$b = 1 - a$

$b = a$

Move $y$ .

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

$b = 1 - a$

$b = a$

Reorder $1$ and $- 1 a y$ .

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

$b = 1 - a$

$b = a$

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

$b = 1 - a$

$b = a$

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

$b = 1 - a, a$