Pre-Algebra Examples

$y = - \frac{23}{2} x - \frac{20}{9}$

Step 1

The standard form of a linear equation is $A x + B y = C$ .

Step 2

Find the LCD of $\frac{23}{2}$ and $\frac{20}{9}$ .

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Step 2.1

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

$2, 9$

Step 2.2

Since $2, 9$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $2, 9$ then find LCM for the variable part .

Step 2.3

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 2.4

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 2.5

$9$ has factors of $3$ and $3$ .

$3 \cdot 3$

Step 2.6

The LCM of $2, 9$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 3 \cdot 3$

Step 2.7

Multiply $2 \cdot 3 \cdot 3$ .

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Step 2.7.1

Multiply $2$ by $3$ .

$6 \cdot 3$

Step 2.7.2

Multiply $6$ by $3$ .

$18$

Step 3

Multiply both sides by $18$ .

$18 y = 18 (- \frac{23}{2} x - \frac{20}{9})$

Step 4

Simplify the right side.

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Step 4.1

Simplify $18 (- \frac{23}{2} x - \frac{20}{9})$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Combine $x$ and $\frac{23}{2}$ .

$18 y = 18 (- \frac{x \cdot 23}{2} - \frac{20}{9})$

Step 4.1.1.2

Move $23$ to the left of $x$ .

$18 y = 18 (- \frac{23 x}{2} - \frac{20}{9})$

Step 4.1.2

Simplify terms.

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Step 4.1.2.1

Apply the distributive property.

$18 y = 18 (- \frac{23 x}{2}) + 18 (- \frac{20}{9})$

Step 4.1.2.2

Cancel the common factor of $2$ .

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Step 4.1.2.2.1

Move the leading negative in $- \frac{23 x}{2}$ into the numerator.

$18 y = 18 \frac{- 23 x}{2} + 18 (- \frac{20}{9})$

Step 4.1.2.2.2

Factor $2$ out of $18$ .

$18 y = 2 (9) \frac{- 23 x}{2} + 18 (- \frac{20}{9})$

Step 4.1.2.2.3

Cancel the common factor.

$18 y = 2 \cdot 9 \frac{- 23 x}{2} + 18 (- \frac{20}{9})$

Step 4.1.2.2.4

Rewrite the expression.

$18 y = 9 (- 23 x) + 18 (- \frac{20}{9})$

Step 4.1.2.3

Multiply $- 23$ by $9$ .

$18 y = - 207 x + 18 (- \frac{20}{9})$

Step 4.1.2.4

Cancel the common factor of $9$ .

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Step 4.1.2.4.1

Move the leading negative in $- \frac{20}{9}$ into the numerator.

$18 y = - 207 x + 18 (\frac{- 20}{9})$

Step 4.1.2.4.2

Factor $9$ out of $18$ .

$18 y = - 207 x + 9 (2) \frac{- 20}{9}$

Step 4.1.2.4.3

Cancel the common factor.

$18 y = - 207 x + 9 \cdot 2 \frac{- 20}{9}$

Step 4.1.2.4.4

Rewrite the expression.

$18 y = - 207 x + 2 \cdot - 20$

Step 4.1.2.5

Multiply $2$ by $- 20$ .

$18 y = - 207 x - 40$

Step 5

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

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Step 5.1

Add $207 x$ to both sides of the equation.

$18 y + 207 x = - 40$

Step 5.2

Reorder $18 y$ and $207 x$ .

$207 x + 18 y = - 40$

Step 6