Solve the following rational expressions.
#1 (4y-3)/8 + (3)/(4y) = (y+3)/2
#2 5/(2x-1) = 1 - (8x-16)/(10x-5)
in each case find the LCD , then multiply each term by that.
I will do the #2
5/(2x-1) = 1 - (8x-16)/(10x-5)
5/(2x-1) = 1 - (8x-16)/(5(2x-1) )
LCD = 5(2x-1)
5/(2x-1)*5(2x-1) = 1*5(2x-1) - (8x-16)/(5(2x-1) )*5(2x-1)
25 = 10x - 5 - (8x-16)
14 = 2x
x = 7
To solve rational expressions, we'll follow these general steps:
Step 1: Clear any denominators.
Step 2: Simplify both sides of the equation.
Step 3: Isolate the variable on one side of the equation.
Step 4: Solve for the variable.
Step 5: Check the solution.
Now, let's solve each rational expression step by step:
#1) (4y-3)/8 + 3/(4y) = (y+3)/2
Step 1: Clear the denominators by multiplying every term by 8 * 4y:
8 * 4y * [(4y-3)/8] + 8 * 4y * [3/(4y)] = 8 * 4y * [(y+3)/2]
Which simplifies to:
4(4y - 3) + 2(3) = (y + 3)(8y)
Step 2: Simplify both sides of the equation:
16y - 12 + 6 = 8y^2 + 24y
16y - 6 = 8y^2 + 24y
Step 3: Isolate the variable:
Rearrange the equation to have all terms on one side:
8y^2 + 24y - 16y + 6 = 0
8y^2 + 8y + 6 = 0
Step 4: Solve for the variable:
Since we are dealing with a quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values:
y = (-(8) ± √((8)^2 - 4(8)(6))) / (2(8))
Simplifying:
y = (-8 ± √(64 - 192))/ 16
y = (-8 ± √(-128))/16
We have a negative value under the square root which means that the equation has no real solutions.
Step 5: Check the solution:
Since we obtained no real solutions, there is nothing to check.
#2) 5/(2x - 1) = 1 - (8x - 16)/(10x - 5)
Step 1: Clear the denominators by multiplying every term by (2x - 1)(10x - 5):
(2x - 1)(10x - 5) * (5/(2x - 1)) = (2x - 1)(10x - 5) * (1 - (8x - 16)/(10x - 5))
Which simplifies to:
5(10x - 5) = (2x - 1)(10x - 5) - (8x - 16)(2x - 1)
Step 2: Simplify both sides of the equation:
50x - 25 = (20x^2 - 10x - 20x + 10) - (16x^2 - 8x - 32x + 16)
50x - 25 = (20x^2 - 10x - 20x + 10) - (16x^2 - 40x + 16)
50x - 25 = (20x^2 - 30x + 10) - (16x^2 - 40x + 16)
50x - 25 = 20x^2 - 30x + 10 - 16x^2 + 40x - 16
Step 3: Isolate the variable:
Rearrange the equation to have all terms on one side:
20x^2 - 30x + 10 - 16x^2 - 10x + 50x - 25 = 0
4x^2 - 30x - 15 = 0
Step 4: Solve for the variable:
Using factoring or the quadratic formula, solve for x.
In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values:
x = (-(30) ± √((30)^2 - 4(4)(-15))) / (2(4))
Simplifying:
x = (-30 ± √(900 + 240)) / 8
x = (-30 ± √1140) / 8
x = (-30 ± √(4 * 285))/8
x = (-30 ± 2√285) / 8
Simplifying further:
x = (-15 ± √285) / 4
So, the solutions are:
x = (-15 + √285) / 4 or x = (-15 - √285) / 4
Step 5: Check the solution:
You can plug these values back into the original equation to check if they satisfy it.
If the equation is satisfied, then the value of x is a valid solution.