Rational Equations Solve:

How do you work it out?

1. x/4 - x+2/3 = 12
(x+2 are in the numerator)

2. 2y/4 - y/7 = y-3/2
(y-3 are in the numerator)

3. p/6 - 2p/5 = 4p-5/15
(4p-5 are in the numerator)

x/4 - (x+2)/3 = 12

multiply by 3*4 or 12 to clear fractions
3 x -4(x+2) = 144

3x - 4 x - 2 = 144

-x = 146

x = - 146

in the second one multiply through by 28 and in the third by 30

1. is -152

3 x -4(x+2) = 144

3x - 4 x - 8 = 144

-x = 152

x = - 152

how would you get 3. if the answer was 2/3

p/6 - 2p/5 = (4p-5)/15

multiply through by 30, the LCM(6,5,15)
5p - 12p = 2(4p-5)
-7p = 8p - 10
15p = 10
p = 2/3

To solve rational equations, follow these steps:

Step 1: Clear the denominators
Multiply through by the least common denominator (LCD) to eliminate the fractions. To find the LCD, you need to factor the denominators and take the highest power of each factor.

Step 2: Simplify
Distribute and combine like terms on both sides of the equation if necessary.

Step 3: Solve for the variable
Isolate the variable on one side of the equation using inverse operations (e.g., addition, subtraction, multiplication, or division).

Let's solve each equation step by step:

1. x/4 - (x+2)/3 = 12

Step 1: Clear the denominators
To find the LCD, factor the denominators:
4 = 2^2
3 = 3^1

The LCD is 2^2 * 3^1 = 12. Multiply every term by 12 to eliminate the fractions:
12 * (x/4) - 12 * ((x+2)/3) = 12 * 12
3x - 4(x+2) = 144

Step 2: Simplify
3x - 4x - 8 = 144
-x - 8 = 144

Step 3: Solve for the variable
Add 8 to both sides:
-x = 144 + 8
-x = 152

Multiply both sides by -1 to isolate x:
x = -152

2. 2y/4 - y/7 = (y-3)/2

Step 1: Clear the denominators
To find the LCD, factor the denominators:
4 = 2^2
7 = 7^1
2 = 2^1

The LCD is 2^2 * 7^1 = 28. Multiply every term by 28 to eliminate the fractions:
28 * (2y/4) - 28 * (y/7) = 28 * ((y-3)/2)
14y - 4y = 14(y-3)

Step 2: Simplify
10y = 14y - 14(3)
10y = 14y - 42

Step 3: Solve for the variable
Subtract 14y from both sides:
10y - 14y = -42
-4y = -42

Divide both sides by -4 to isolate y:
y = (-42) / (-4)
y = 10.5

3. p/6 - (2p/5) = (4p-5)/15

Step 1: Clear the denominators
To find the LCD, factor the denominators:
6 = 2 * 3
5 = 5^1
15 = 3 * 5

The LCD is 2 * 3 * 5 = 30. Multiply every term by 30 to eliminate the fractions:
30 * (p/6) - 30 * (2p/5) = 30 * ((4p-5)/15)
5p - 12p = 2(4p-5)

Step 2: Simplify
-7p = 8p - 10

Step 3: Solve for the variable
Subtract 8p from both sides:
-7p - 8p = -10
-15p = -10

Divide both sides by -15 to isolate p:
p = (-10) / (-15)
p = 2/3

Therefore, the solutions for the given rational equations are:
1. x = -152
2. y = 10.5
3. p = 2/3