Linear Algebra Examples

$\frac{a - b}{a + b + c} = \frac{a - 2 b + c}{a - b}$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$a + b + c, a - b$

Step 1.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.4

The LCM of $1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 1.5

The factor for $a + b + c$ is $a + b + c$ itself.

$(a + b + c) = a + b + c$

$(a + b + c)$ occurs $1$ time.

Step 1.6

The factor for $a - b$ is $a - b$ itself.

$(a - b) = a - b$

$(a - b)$ occurs $1$ time.

Step 1.7

The LCM of $a + b + c, a - b$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(a + b + c) (a - b)$

Step 2

Multiply each term in $\frac{a - b}{a + b + c} = \frac{a - 2 b + c}{a - b}$ by $(a + b + c) (a - b)$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{a - b}{a + b + c} = \frac{a - 2 b + c}{a - b}$ by $(a + b + c) (a - b)$ .

$\frac{a - b}{a + b + c} ((a + b + c) (a - b)) = \frac{a - 2 b + c}{a - b} ((a + b + c) (a - b))$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $a + b + c$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{a - b}{a + b + c} ((a + b + c) (a - b)) = \frac{a - 2 b + c}{a - b} ((a + b + c) (a - b))$

Step 2.2.1.2

Rewrite the expression.

$(a - b) (a - b) = \frac{a - 2 b + c}{a - b} ((a + b + c) (a - b))$

Step 2.2.2

Raise $a - b$ to the power of $1$ .

${(a - b)}^{1} (a - b) = \frac{a - 2 b + c}{a - b} ((a + b + c) (a - b))$

Step 2.2.3

Raise $a - b$ to the power of $1$ .

${(a - b)}^{1} {(a - b)}^{1} = \frac{a - 2 b + c}{a - b} ((a + b + c) (a - b))$

Step 2.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(a - b)}^{1 + 1} = \frac{a - 2 b + c}{a - b} ((a + b + c) (a - b))$

Step 2.2.5

Add $1$ and $1$ .

${(a - b)}^{2} = \frac{a - 2 b + c}{a - b} ((a + b + c) (a - b))$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $a - b$ .

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Step 2.3.1.1

Factor $a - b$ out of $(a + b + c) (a - b)$ .

${(a - b)}^{2} = \frac{a - 2 b + c}{a - b} ((a - b) (a + b + c))$

Step 2.3.1.2

Cancel the common factor.

${(a - b)}^{2} = \frac{a - 2 b + c}{a - b} ((a - b) (a + b + c))$

Step 2.3.1.3

Rewrite the expression.

${(a - b)}^{2} = (a - 2 b + c) (a + b + c)$

Step 2.3.2

Expand $(a - 2 b + c) (a + b + c)$ by multiplying each term in the first expression by each term in the second expression.

${(a - b)}^{2} = a \cdot a + a b + a c - 2 b a - 2 b \cdot b - 2 b c + c a + c b + c \cdot c$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Multiply $a$ by $a$ .

${(a - b)}^{2} = a^{2} + a b + a c - 2 b a - 2 b \cdot b - 2 b c + c a + c b + c \cdot c$

Step 2.3.3.2

Multiply $b$ by $b$ by adding the exponents.

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Step 2.3.3.2.1

Move $b$ .

${(a - b)}^{2} = a^{2} + a b + a c - 2 b a - 2 (b \cdot b) - 2 b c + c a + c b + c \cdot c$

Step 2.3.3.2.2

Multiply $b$ by $b$ .

${(a - b)}^{2} = a^{2} + a b + a c - 2 b a - 2 b^{2} - 2 b c + c a + c b + c \cdot c$

Step 2.3.3.3

Multiply $c$ by $c$ .

${(a - b)}^{2} = a^{2} + a b + a c - 2 b a - 2 b^{2} - 2 b c + c a + c b + c^{2}$

Step 2.3.4

Subtract $2 b a$ from $a b$ .

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Step 2.3.4.1

Move $b$ .

${(a - b)}^{2} = a^{2} + a b - 2 a b + a c - 2 b^{2} - 2 b c + c a + c b + c^{2}$

Step 2.3.4.2

Subtract $2 a b$ from $a b$ .

${(a - b)}^{2} = a^{2} - a b + a c - 2 b^{2} - 2 b c + c a + c b + c^{2}$

Step 2.3.5

Add $a c$ and $c a$ .

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Step 2.3.5.1

Reorder $c$ and $a$ .

${(a - b)}^{2} = a^{2} - a b - 2 b^{2} - 2 b c + a c + a c + c b + c^{2}$

Step 2.3.5.2

Add $a c$ and $a c$ .

${(a - b)}^{2} = a^{2} - a b - 2 b^{2} - 2 b c + 2 a c + c b + c^{2}$

Step 2.3.6

Add $- 2 b c$ and $c b$ .

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Step 2.3.6.1

Reorder $c$ and $b$ .

${(a - b)}^{2} = a^{2} - a b - 2 b^{2} - 2 b c + b c + 2 a c + c^{2}$

Step 2.3.6.2

Add $- 2 b c$ and $b c$ .

${(a - b)}^{2} = a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2}$

Step 3

Solve the equation.

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Step 3.1

Since $a$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = {(a - b)}^{2}$

Step 3.2

Simplify ${(a - b)}^{2}$ .

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Step 3.2.1

Rewrite.

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = 0 + 0 + {(a - b)}^{2}$

Step 3.2.2

Rewrite ${(a - b)}^{2}$ as $(a - b) (a - b)$ .

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = (a - b) (a - b)$

Step 3.2.3

Expand $(a - b) (a - b)$ using the FOIL Method.

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Step 3.2.3.1

Apply the distributive property.

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = a (a - b) - b (a - b)$

Step 3.2.3.2

Apply the distributive property.

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = a \cdot a + a (- b) - b (a - b)$

Step 3.2.3.3

Apply the distributive property.

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = a \cdot a + a (- b) - b a - b (- b)$

Step 3.2.4

Simplify and combine like terms.

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

Multiply $a$ by $a$ .

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = a^{2} + a (- b) - b a - b (- b)$

Step 3.2.4.1.2

Rewrite using the commutative property of multiplication.

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = a^{2} - a b - b a - b (- b)$

Step 3.2.4.1.3

Rewrite using the commutative property of multiplication.

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = a^{2} - a b - b a - 1 \cdot - 1 b \cdot b$

Step 3.2.4.1.4

Multiply $b$ by $b$ by adding the exponents.

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Step 3.2.4.1.4.1

Move $b$ .

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = a^{2} - a b - b a - 1 \cdot - 1 (b \cdot b)$

Step 3.2.4.1.4.2

Multiply $b$ by $b$ .

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = a^{2} - a b - b a - 1 \cdot - 1 b^{2}$

Step 3.2.4.1.5

Multiply $- 1$ by $- 1$ .

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = a^{2} - a b - b a + 1 b^{2}$

Step 3.2.4.1.6

Multiply $b^{2}$ by $1$ .

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = a^{2} - a b - b a + b^{2}$

Step 3.2.4.2

Subtract $b a$ from $- a b$ .

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Step 3.2.4.2.1

Move $b$ .

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = a^{2} - a b - 1 a b + b^{2}$

Step 3.2.4.2.2

Subtract $a b$ from $- a b$ .

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} = a^{2} - 2 a b + b^{2}$

Step 3.3

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

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Step 3.3.1

Subtract $a^{2}$ from both sides of the equation.

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} - a^{2} = - 2 a b + b^{2}$

Step 3.3.2

Add $2 a b$ to both sides of the equation.

$a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} - a^{2} + 2 a b = b^{2}$

Step 3.3.3

Combine the opposite terms in $a^{2} - a b - 2 b^{2} - b c + 2 a c + c^{2} - a^{2} + 2 a b$ .

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Step 3.3.3.1

Subtract $a^{2}$ from $a^{2}$ .

$- a b - 2 b^{2} - b c + 2 a c + c^{2} + 0 + 2 a b = b^{2}$

Step 3.3.3.2

Add $- a b - 2 b^{2} - b c + 2 a c + c^{2}$ and $0$ .

$- a b - 2 b^{2} - b c + 2 a c + c^{2} + 2 a b = b^{2}$

Step 3.3.4

Add $- a b$ and $2 a b$ .

$- 2 b^{2} - b c + 2 a c + c^{2} + 1 a b = b^{2}$

Step 3.3.5

Multiply $a$ by $1$ .

$- 2 b^{2} - b c + 2 a c + c^{2} + a b = b^{2}$

Step 3.4

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

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Step 3.4.1

Add $2 b^{2}$ to both sides of the equation.

$- b c + 2 a c + c^{2} + a b = b^{2} + 2 b^{2}$

Step 3.4.2

Add $b c$ to both sides of the equation.

$2 a c + c^{2} + a b = b^{2} + 2 b^{2} + b c$

Step 3.4.3

Subtract $c^{2}$ from both sides of the equation.

$2 a c + a b = b^{2} + 2 b^{2} + b c - c^{2}$

Step 3.4.4

Add $b^{2}$ and $2 b^{2}$ .

$2 a c + a b = 3 b^{2} + b c - c^{2}$

Step 3.5

Factor $a$ out of $2 a c + a b$ .

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Step 3.5.1

Factor $a$ out of $2 a c$ .

$a (2 c) + a b = 3 b^{2} + b c - c^{2}$

Step 3.5.2

Factor $a$ out of $a b$ .

$a (2 c) + a (b) = 3 b^{2} + b c - c^{2}$

Step 3.5.3

Factor $a$ out of $a (2 c) + a (b)$ .

$a (2 c + b) = 3 b^{2} + b c - c^{2}$

Step 3.6

Divide each term in $a (2 c + b) = 3 b^{2} + b c - c^{2}$ by $2 c + b$ and simplify.

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Step 3.6.1

Divide each term in $a (2 c + b) = 3 b^{2} + b c - c^{2}$ by $2 c + b$ .

$\frac{a (2 c + b)}{2 c + b} = \frac{3 b^{2}}{2 c + b} + \frac{b c}{2 c + b} + \frac{- c^{2}}{2 c + b}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $2 c + b$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{a (2 c + b)}{2 c + b} = \frac{3 b^{2}}{2 c + b} + \frac{b c}{2 c + b} + \frac{- c^{2}}{2 c + b}$

Step 3.6.2.1.2

Divide $a$ by $1$ .

$a = \frac{3 b^{2}}{2 c + b} + \frac{b c}{2 c + b} + \frac{- c^{2}}{2 c + b}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Move the negative in front of the fraction.

$a = \frac{3 b^{2}}{2 c + b} + \frac{b c}{2 c + b} - \frac{c^{2}}{2 c + b}$

Step 3.6.3.2

Combine the numerators over the common denominator.

$a = \frac{3 b^{2} + b c}{2 c + b} - \frac{c^{2}}{2 c + b}$

Step 3.6.3.3

Combine the numerators over the common denominator.

$a = \frac{3 b^{2} + b c - c^{2}}{2 c + b}$

Step 4

Set the denominator in $\frac{3 b^{2} + b c - c^{2}}{2 c + b}$ equal to $0$ to find where the expression is undefined.

$2 c + b = 0$

Step 5

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

$b = - 2 c$

Step 6

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${b | b \in ℝ}$