Solve for x: [tex]\frac{1}{3} x-\frac{1}{2}=\frac{3}{4} x[/tex]
Answer (2)
Multiply both sides of the equation by the least common multiple of the denominators to eliminate fractions: 12 ( 3 1 x − 2 1 ) = 12 ( 4 3 x ) .
Simplify the equation: 4 x − 6 = 9 x .
Isolate the x terms: − 6 = 5 x .
Solve for x: x = − 5 6 .
Explanation
Understanding the Problem We are given the equation 3 1 x − 2 1 = 4 3 x Our goal is to solve for x .
Eliminating Fractions To eliminate the fractions, we multiply both sides of the equation by the least common multiple (LCM) of the denominators 3, 2, and 4. The LCM of 3, 2, and 4 is 12. Multiplying both sides by 12, we get: 12 × ( 3 1 x − 2 1 ) = 12 × ( 4 3 x ) 12 × 3 1 x − 12 × 2 1 = 12 × 4 3 x 4 x − 6 = 9 x
Isolating x Now, we want to isolate the x terms on one side of the equation and the constant terms on the other side. We can subtract 4 x from both sides: 4 x − 6 − 4 x = 9 x − 4 x − 6 = 5 x
Solving for x Finally, we solve for x by dividing both sides by 5: 5 − 6 = 5 5 x x = − 5 6
Final Answer Therefore, the solution to the equation is x = − 5 6 .
Examples
In physics, you might use linear equations like this to model the motion of an object with constant velocity. For example, if you have a car moving at a certain speed and you want to determine when it will reach a specific point, you could set up an equation similar to this one. Solving for the variable would tell you the time at which the car reaches that point. Understanding how to solve these equations is crucial for making predictions and solving problems in various scientific and engineering fields.
To solve the equation 3 1 x − 2 1 = 4 3 x , we multiply through by 12 to eliminate fractions, resulting in 4 x − 6 = 9 x . After isolating x and simplifying, we find that x = − 5 6 .
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