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Solution - Absolute value equations

Exact form:

y = 0, 0

y=0 , 0

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| \frac{2}{5} y | = | \frac{4}{5} y |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| \frac{2}{5} y \| = \| \frac{4}{5} y \|$
$x = + y$	$(\frac{2}{5} y) = (\frac{4}{5} y)$
$x = - y$	$(\frac{2}{5} y) = - (\frac{4}{5} y)$
$+ x = y$	$(\frac{2}{5} y) = (\frac{4}{5} y)$
$- x = y$	$- (\frac{2}{5} y) = (\frac{4}{5} y)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$\| \frac{2}{5} y \| = \| \frac{4}{5} y \|$
$x = + y$ , $+ x = y$	$(\frac{2}{5} y) = (\frac{4}{5} y)$
$x = - y$ , $- x = y$	$(\frac{2}{5} y) = - (\frac{4}{5} y)$

2. Solve the two equations for y

7 additional steps

$\frac{2}{5} \cdot y = \frac{4}{5} y$

Subtract from both sides:

$(\frac{2}{5} y) - \frac{4}{5} \cdot y = (\frac{4}{5} y) - \frac{4}{5} y$

Combine the fractions:

$\frac{(2 - 4)}{5} \cdot y = (\frac{4}{5} \cdot y) - \frac{4}{5} y$

Combine the numerators:

$\frac{- 2}{5} \cdot y = (\frac{4}{5} \cdot y) - \frac{4}{5} y$

Combine the fractions:

$\frac{- 2}{5} \cdot y = \frac{(4 - 4)}{5} y$

Combine the numerators:

$\frac{- 2}{5} \cdot y = \frac{0}{5} y$

Reduce the zero numerator:

$\frac{- 2}{5} y = 0 y$

Simplify the arithmetic:

$\frac{- 2}{5} y = 0$

Divide both sides by the coefficient:

$y = 0$

10 additional steps

$\frac{2}{5} \cdot y = \frac{- 4}{5} y$

Multiply both sides by inverse fraction :

$(\frac{2}{5} y) \cdot \frac{5}{2} = (\frac{- 4}{5} y) \cdot \frac{5}{2}$

Group like terms:

$(\frac{2}{5} \cdot \frac{5}{2}) y = (\frac{- 4}{5} y) \cdot \frac{5}{2}$

Multiply the coefficients:

$\frac{(2 \cdot 5)}{(5 \cdot 2)} \cdot y = (\frac{- 4}{5} y) \cdot \frac{5}{2}$

Simplify the fraction:

$y = (\frac{- 4}{5} y) \cdot \frac{5}{2}$

Group like terms:

$y = (\frac{- 4}{5} \cdot \frac{5}{2}) y$

Multiply the coefficients:

$y = \frac{(- 4 \cdot 5)}{(5 \cdot 2)} y$

Simplify the arithmetic:

$y = - 2 y$

Add to both sides:

$y + 2 y = (- 2 y) + 2 y$

Simplify the arithmetic:

$3 y = (- 2 y) + 2 y$

Simplify the arithmetic:

$3 y = 0$

Divide both sides by the coefficient:

$y = 0$

3. List the solutions

$y = 0, 0$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | \frac{2}{5} y |$
$y = | \frac{4}{5} y |$
The equation is true where the two lines cross.

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Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for y

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for y

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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