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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
$| 4 y | = | 202 y |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 y \| = \| 202 y \|$
$x = + y$	$(4 y) = (202 y)$
$x = - y$	$(4 y) = - (202 y)$
$+ x = y$	$(4 y) = (202 y)$
$- x = y$	$- (4 y) = (202 y)$

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:

$\| x \| = \| y \|$	$\| 4 y \| = \| 202 y \|$
$x = + y$ , $+ x = y$	$(4 y) = (202 y)$
$x = - y$ , $- x = y$	$(4 y) = - (202 y)$

2. Solve the two equations for y

3 additional steps

$4 y = 202 y$

Subtract from both sides:

$(4 y) - 202 y = (202 y) - 202 y$

Simplify the arithmetic:

$- 198 y = (202 y) - 202 y$

Simplify the arithmetic:

$- 198 y = 0$

Divide both sides by the coefficient:

$y = 0$

12 additional steps

$4 y = - 202 y$

Divide both sides by :

$\frac{(4 y)}{4} = \frac{(- 202 y)}{4}$

Simplify the fraction:

$y = \frac{(- 202 y)}{4}$

Simplify the fraction:

$y = \frac{- 101}{2} y$

Add to both sides:

$y + \frac{101}{2} \cdot y = (\frac{- 101}{2} y) + \frac{101}{2} y$

Group the coefficients:

$(1 + \frac{101}{2}) y = (\frac{- 101}{2} \cdot y) + \frac{101}{2} y$

Convert the integer into a fraction:

$(\frac{2}{2} + \frac{101}{2}) y = (\frac{- 101}{2} \cdot y) + \frac{101}{2} y$

Combine the fractions:

$\frac{(2 + 101)}{2} \cdot y = (\frac{- 101}{2} \cdot y) + \frac{101}{2} y$

Combine the numerators:

$\frac{103}{2} \cdot y = (\frac{- 101}{2} \cdot y) + \frac{101}{2} y$

Combine the fractions:

$\frac{103}{2} \cdot y = \frac{(- 101 + 101)}{2} y$

Combine the numerators:

$\frac{103}{2} \cdot y = \frac{0}{2} y$

Reduce the zero numerator:

$\frac{103}{2} y = 0 y$

Simplify the arithmetic:

$\frac{103}{2} y = 0$

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:
$y = | 4 y |$
$y = | 202 y |$
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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