Calculus Examples

$8 x - 2 y = 8 y - 2 x$ , $8 x + 2 x = 8 y + 2 y$

Step 1

Simplify the right side.

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Step 1.1

Reorder $8 y$ and $- 2 x$ .

$8 x - 2 y = - 2 x + 8 y$

$8 x + 2 x = 8 y + 2 y$

$8 x - 2 y = - 2 x + 8 y$

$8 x + 2 x = 8 y + 2 y$

Step 2

Simplify the left side.

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Step 2.1

Add $8 x$ and $2 x$ .

$10 x = 8 y + 2 y$

$8 x - 2 y = - 2 x + 8 y$

$10 x = 8 y + 2 y$

$8 x - 2 y = - 2 x + 8 y$

Step 3

Simplify the right side.

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Step 3.1

Add $8 y$ and $2 y$ .

$10 x = 10 y$

$8 x - 2 y = - 2 x + 8 y$

$10 x = 10 y$

$8 x - 2 y = - 2 x + 8 y$

Step 4

Move all terms containing variables to the left.

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Step 4.1

Move all terms containing variables to the left side of the equation.

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Step 4.1.1

Add $2 x$ to both sides of the equation.

$8 x - 2 y + 2 x = 8 y, 10 x = 10 y$

Step 4.1.2

Subtract $8 y$ from both sides of the equation.

$8 x - 2 y + 2 x - 8 y = 0, 10 x = 10 y$

Step 4.1.3

Add $8 x$ and $2 x$ .

$10 x - 2 y - 8 y = 0, 10 x = 10 y$

Step 4.1.4

Subtract $8 y$ from $- 2 y$ .

$10 x - 10 y = 0, 10 x = 10 y$

Step 4.2

Subtract $10 y$ from both sides of the equation.

$10 x - 10 y = 0, 10 x - 10 y = 0$

Step 5

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

$(- 1) \cdot (10 x - 10 y) = (- 1) (0)$

$10 x - 10 y = 0$

Step 6

Simplify.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $(- 1) \cdot (10 x - 10 y)$ .

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Step 6.1.1.1

Apply the distributive property.

$- 1 (10 x) - 1 (- 10 y) = (- 1) (0)$

$10 x - 10 y = 0$

Step 6.1.1.2

Multiply.

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Step 6.1.1.2.1

Multiply $10$ by $- 1$ .

$- 10 x - 1 (- 10 y) = (- 1) (0)$

$10 x - 10 y = 0$

Step 6.1.1.2.2

Multiply $- 10$ by $- 1$ .

$- 10 x + 10 y = (- 1) (0)$

$10 x - 10 y = 0$

$- 10 x + 10 y = (- 1) (0)$

$10 x - 10 y = 0$

$- 10 x + 10 y = (- 1) (0)$

$10 x - 10 y = 0$

$- 10 x + 10 y = (- 1) (0)$

$10 x - 10 y = 0$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $- 1$ by $0$ .

$- 10 x + 10 y = 0$

$10 x - 10 y = 0$

$- 10 x + 10 y = 0$

$10 x - 10 y = 0$

$- 10 x + 10 y = 0$

$10 x - 10 y = 0$

Step 7

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

	$-$	$1$	$0$	$x$	$+$	$1$	$0$	$y$	$=$	$0$
$+$		$1$	$0$	$x$	$-$	$1$	$0$	$y$	$=$	$0$
								$0$	$=$	$0$

Step 8

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

Infinite number of solutions

Step 9

Solve one of the equations for $y$ .

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Step 9.1

Add $10 x$ to both sides of the equation.

$10 y = 10 x$

Step 9.2

Divide each term in $10 y = 10 x$ by $10$ and simplify.

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Step 9.2.1

Divide each term in $10 y = 10 x$ by $10$ .

$\frac{10 y}{10} = \frac{10 x}{10}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $10$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{10 y}{10} = \frac{10 x}{10}$

Step 9.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{10 x}{10}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Cancel the common factor of $10$ .

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Step 9.2.3.1.1

Cancel the common factor.

$y = \frac{10 x}{10}$

Step 9.2.3.1.2

Divide $x$ by $1$ .

$y = x$

Step 10

The solution is the set of ordered pairs that make $y = x$ true.

$(x, x)$

Step 11

The result can be shown in multiple forms.

Point Form:

$(x, x)$

Equation Form:

$x = x, y = x$

Step 12

	$-$	$1$	$0$	$x$	$+$	$1$	$0$	$y$	$=$	$0$
$+$		$1$	$0$	$x$	$-$	$1$	$0$	$y$	$=$	$0$
								$0$	$=$	$0$

	$-$	$1$	$0$	$x$	$+$	$1$	$0$	$y$	$=$	$0$
$+$		$1$	$0$	$x$	$-$	$1$	$0$	$y$	$=$	$0$
								$0$	$=$	$0$

	$-$	$1$	$0$	$x$	$+$	$1$	$0$	$y$	$=$	$0$
$+$		$1$	$0$	$x$	$-$	$1$	$0$	$y$	$=$	$0$
								$0$	$=$	$0$