5x+6/3x-9=7/3

ambiguous.

did you mean the way you typed it, or
(5x+6)/(3x-9)=7/3
or
5x+6/(3x) -9=7/3
or ..
?

The first example

(5x+6)/(3x-9)=7/3

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clear fractions, or cross-multiply

3(5x+6) = 7(3x-9)
15x+18 = 21x-63
...

the LCD is 3(3x-9), so let's multiply both sides by that

3(5x+6) = 7(3x-9)
this is often called "cross-multiplying"

15x + 18 = 21x - 63
-6x = - 81
x = -81/-6 = 13.5

To solve the equation 5x + (6 / 3x - 9) = 7 / 3, we need to first simplify both sides of the equation. Let's break down the steps to solve it.

Step 1: Simplify the equation by finding a common denominator for the fractions.
The common denominator for the fractions on both sides of the equation is 3x - 9, since it's the denominator of the fraction on the left side. Multiply both the numerator and denominator of the fraction on the right side by (3x - 9) to get a common denominator.

5x + (6 / 3x - 9) = (7 / 3)
5x + (6 * (3x - 9)) / (3x - 9) = (7 / 3)

Simplifying the right side:

5x + (18x - 54) / (3x - 9) = 7 / 3

Step 2: Combine like terms on the left side.
Combine the 5x and the 18x terms:

5x + 18x - 54 / (3x - 9) = 7 / 3

23x - 54 / (3x - 9) = 7 / 3

Step 3: Multiply both sides of the equation by the denominators to eliminate fractions.
Multiply both sides by (3x - 9):

(3x - 9)(23x - 54) / (3x - 9) = (3x - 9)(7 / 3)

Simplifying the left side:

23x - 54 = 7(3x - 9) / 3

Step 4: Distribute on the right side:
Distribute 7/3 to both 3x and -9:

23x - 54 = (7(3x) - 7(9)) / 3

23x - 54 = (21x - 63) / 3

Step 5: Multiply both sides by 3 to eliminate the fraction.
Multiply both sides by 3:

3(23x - 54) = 3((21x - 63) / 3)

Simplifying both sides:

69x - 162 = 21x - 63

Step 6: Isolate the x variable on one side of the equation.
Move the constant terms to the right side by adding 162 to both sides:

69x - 162 + 162 = 21x - 63 + 162

Simplifying both sides:

69x = 21x + 99

Step 7: Move the 21x term to the left side by subtracting 21x from both sides:

69x - 21x = 21x + 99 - 21x

Simplifying both sides:

48x = 99

Step 8: Solve for x by dividing both sides by 48:

(48x) / 48 = (99) / 48

Simplifying both sides:

x = 99 / 48

This is the final solution for x, which can be simplified further if needed.