Algebra Examples

$a x - b y = c$ $a x - b y = c$

Step 1

Move all terms containing variables to the left.

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Step 1.1

Subtract $c$ from both sides of the equation.

$a x - b y - c = 0, a x - b y = c$

Step 1.2

Subtract $c$ from both sides of the equation.

$a x - b y - c = 0, a x - b y - c = 0$

Step 2

Multiply each equation by the value that makes the coefficients of $a$ opposite.

$(- 1) \cdot (a x - b y - c) = (- 1) (0)$

$a x - b y - c = 0$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $(- 1) \cdot (a x - b y - c)$ .

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Step 3.1.1.1

Apply the distributive property.

$- 1 (a x) - 1 (- b y) - 1 (- c) = (- 1) (0)$

$a x - b y - c = 0$

Step 3.1.1.2

Simplify.

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Step 3.1.1.2.1

Rewrite $- 1 (a x)$ as $- (a x)$ .

$- (a x) - 1 (- b y) - 1 (- c) = (- 1) (0)$

$a x - b y - c = 0$

Step 3.1.1.2.2

Multiply $- 1 (- b y)$ .

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Step 3.1.1.2.2.1

Multiply $- 1$ by $- 1$ .

$- (a x) + 1 (b y) - 1 (- c) = (- 1) (0)$

$a x - b y - c = 0$

Step 3.1.1.2.2.2

Multiply $b$ by $1$ .

$- (a x) + b y - 1 (- c) = (- 1) (0)$

$a x - b y - c = 0$

$- (a x) + b y - 1 (- c) = (- 1) (0)$

$a x - b y - c = 0$

Step 3.1.1.2.3

Multiply $- 1 (- c)$ .

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Step 3.1.1.2.3.1

Multiply $- 1$ by $- 1$ .

$- (a x) + b y + 1 c = (- 1) (0)$

$a x - b y - c = 0$

Step 3.1.1.2.3.2

Multiply $c$ by $1$ .

$- (a x) + b y + c = (- 1) (0)$

$a x - b y - c = 0$

$- (a x) + b y + c = (- 1) (0)$

$a x - b y - c = 0$

$- a x + b y + c = (- 1) (0)$

$a x - b y - c = 0$

$- a x + b y + c = (- 1) (0)$

$a x - b y - c = 0$

$- a x + b y + c = (- 1) (0)$

$a x - b y - c = 0$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$- a x + b y + c = 0$

$a x - b y - c = 0$

$- a x + b y + c = 0$

$a x - b y - c = 0$

$- a x + b y + c = 0$

$a x - b y - c = 0$

Step 4

Add the two equations together to eliminate $a$ from the system.

$-$

$a$

$x$

$+$

$b$

$y$

$+$

$c$

$=$

$0$

$+$

$a$

$x$

$-$

$b$

$y$

$-$

$c$

$=$

$0$

$=$

$0$

Step 5

Since $0 = 0$ , the equations intersect at an infinite number of points.

Infinite number of solutions

Step 6

Solve one of the equations for $x$ .

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Step 6.1

Move all terms not containing $x$ to the right side of the equation.

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Step 6.1.1

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

$- a x + c = - b y$

Step 6.1.2

Subtract $c$ from both sides of the equation.

$- a x = - b y - c$

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.2.2.2

Cancel the common factor of $a$ .

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Step 6.2.2.2.1

Cancel the common factor.

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

Step 6.2.2.2.2

Divide $x$ by $1$ .

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Simplify each term.

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Step 6.2.3.1.1

Dividing two negative values results in a positive value.

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

Step 6.2.3.1.2

Dividing two negative values results in a positive value.

$x = \frac{b y}{a} + \frac{c}{a}$

Step 7

The solution is the set of ordered pairs that make $x = \frac{b y}{a} + \frac{c}{a}$ true.

$(\frac{b y}{a} + \frac{c}{a}, a)$