Algebra Examples

$| y - 3 | = 2 y - 10$

Step 1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y - 3 = \pm (2 y - 10)$

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.1

First, use the positive value of the $\pm$ to find the first solution.

$y - 3 = 2 y - 10$

Step 2.2

Move all terms containing $y$ to the left side of the equation.

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Step 2.2.1

Subtract $2 y$ from both sides of the equation.

$y - 3 - 2 y = - 10$

Step 2.2.2

Subtract $2 y$ from $y$ .

$- y - 3 = - 10$

Step 2.3

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

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Step 2.3.1

Add $3$ to both sides of the equation.

$- y = - 10 + 3$

Step 2.3.2

Add $- 10$ and $3$ .

$- y = - 7$

Step 2.4

Divide each term in $- y = - 7$ by $- 1$ and simplify.

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Step 2.4.1

Divide each term in $- y = - 7$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 7}{- 1}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 7}{- 1}$

Step 2.4.2.2

Divide $y$ by $1$ .

$y = \frac{- 7}{- 1}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $- 7$ by $- 1$ .

$y = 7$

Step 2.5

Next, use the negative value of the $\pm$ to find the second solution.

$y - 3 = - (2 y - 10)$

Step 2.6

Simplify $- (2 y - 10)$ .

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Step 2.6.1

Rewrite.

$y - 3 = 0 + 0 - (2 y - 10)$

Step 2.6.2

Simplify by adding zeros.

$y - 3 = - (2 y - 10)$

Step 2.6.3

Apply the distributive property.

$y - 3 = - (2 y) - - 10$

Step 2.6.4

Multiply.

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Step 2.6.4.1

Multiply $2$ by $- 1$ .

$y - 3 = - 2 y - - 10$

Step 2.6.4.2

Multiply $- 1$ by $- 10$ .

$y - 3 = - 2 y + 10$

Step 2.7

Move all terms containing $y$ to the left side of the equation.

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Step 2.7.1

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

$y - 3 + 2 y = 10$

Step 2.7.2

Add $y$ and $2 y$ .

$3 y - 3 = 10$

Step 2.8

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

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Step 2.8.1

Add $3$ to both sides of the equation.

$3 y = 10 + 3$

Step 2.8.2

Add $10$ and $3$ .

$3 y = 13$

Step 2.9

Divide each term in $3 y = 13$ by $3$ and simplify.

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Step 2.9.1

Divide each term in $3 y = 13$ by $3$ .

$\frac{3 y}{3} = \frac{13}{3}$

Step 2.9.2

Simplify the left side.

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Step 2.9.2.1

Cancel the common factor of $3$ .

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Step 2.9.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{13}{3}$

Step 2.9.2.1.2

Divide $y$ by $1$ .

$y = \frac{13}{3}$

Step 2.10

The complete solution is the result of both the positive and negative portions of the solution.

$y = 7, \frac{13}{3}$

Step 3

Exclude the solutions that do not make $| y - 3 | = 2 y - 10$ true.

$y = 7$