What is the value of $x$ in the equation $\frac{1}{2} x-\frac{3}{4}=\frac{3}{8}-\frac{5}{8} x $?
Answer (2)
The value of x in the equation 2 1 x − 4 3 = 8 3 − 8 5 x is 1 . We solve by eliminating fractions, simplifying, and isolating x . Ultimately, we find that x = 1 .
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Multiply both sides of the equation by 8 to eliminate fractions: 8 ( 2 1 x − 4 3 ) = 8 ( 8 3 − 8 5 x ) .
Simplify the equation: 4 x − 6 = 3 − 5 x .
Isolate x by adding 5 x and 6 to both sides: 9 x = 9 .
Solve for x by dividing both sides by 9: x = 1 , so the final answer is 1 .
Explanation
Problem Analysis We are given the equation 2 1 x − 4 3 = 8 3 − 8 5 x and we want to find the value of x that satisfies this equation.
Eliminating Fractions To solve for x , we first want to eliminate the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which are 2, 4, and 8. The LCM of 2, 4, and 8 is 8. So, we multiply both sides of the equation by 8:
8 ( 2 1 x − 4 3 ) = 8 ( 8 3 − 8 5 x )
Simplifying the Equation Now, distribute the 8 on both sides of the equation:
8 × 2 1 x − 8 × 4 3 = 8 × 8 3 − 8 × 8 5 x
This simplifies to:
4 x − 6 = 3 − 5 x
Isolating x Next, we want to isolate x on one side of the equation. Add 5 x to both sides:
4 x + 5 x − 6 = 3 − 5 x + 5 x
This simplifies to:
9 x − 6 = 3
Further Isolating x Now, add 6 to both sides of the equation:
9 x − 6 + 6 = 3 + 6
This simplifies to:
9 x = 9
Solving for x Finally, divide both sides by 9 to solve for x :
9 9 x = 9 9
This simplifies to:
x = 1
So, the value of x that satisfies the equation is 1.
Final Answer Therefore, the value of x in the equation 2 1 x − 4 3 = 8 3 − 8 5 x is x = 1 .
Examples
Imagine you're baking a cake and need to adjust a recipe. The original recipe calls for certain fractions of ingredients, but you want to scale it down. Solving linear equations with fractions, like the one we just tackled, helps you determine the exact amounts of each ingredient needed to maintain the cake's deliciousness. This skill is also useful in managing finances, calculating discounts, or determining proportions in various real-life scenarios where precision is key.
