Solving the Equation 2/3x - 3/12 = 4/5 - (7/x - 2)
In this article, we will solve the equation 2/3x - 3/12 = 4/5 - (7/x - 2). This equation involves fractions and variables, making it a bit more challenging than a simple linear equation.
Step 1: Simplify the Equation
First, let's simplify the equation by combining like terms:
2/3x - 3/12 = 4/5 - (7/x - 2)
To simplify the right-hand side of the equation, we need to follow the order of operations (PEMDAS):
- Evaluate the expression inside the parentheses: 7/x - 2 = (7 - 2x)/x
- Simplify the fraction: (7 - 2x)/x = 7/x - 2/x
- Rewrite the equation with the simplified expression: 2/3x - 3/12 = 4/5 - 7/x + 2/x
Step 2: Get a Common Denominator
Next, we need to get a common denominator for all the fractions in the equation. The least common multiple (LCM) of 3, 12, 5, and x is 60x. We can multiply each fraction by the appropriate factor to get a common denominator:
(40x/60x)x - (15/60x) = (48/60x)/5 - (28/60x)/x + (12/60x)/x
Step 3: Simplify and Combine Like Terms
Now, we can simplify the equation by combining like terms:
40x - 15 = 48/5 - 28/x + 12/x
Step 4: Solve for x
To solve for x, we can add 15 to both sides of the equation:
40x = 48/5 + 12/x - 28/x + 15
Next, we can multiply both sides of the equation by x to eliminate the fractions:
40x^2 = 48x/5 + 12 - 28 + 15x
Step 5: Solve the Quadratic Equation
Now, we have a quadratic equation in terms of x. We can simplify the equation by combining like terms:
40x^2 - 15x - 48/5 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 40, b = -15, and c = -48/5. Plugging these values into the formula, we get:
x = (15 ± √((-15)^2 - 4(40)(-48/5))) / 80
Simplifying the equation, we get two possible values for x:
x = 1.25 or x = -0.6
Conclusion
In this article, we have solved the equation 2/3x - 3/12 = 4/5 - (7/x - 2) using algebraic manipulations and the quadratic formula. The solutions to the equation are x = 1.25 and x = -0.6.
