$-2 x=-\frac{1}{4}+\frac{1}{2} x+5$ to justify the solution. What value of $x$ is the solution?
$x=\square$
Answer (1)
Combine constant terms: Rewrite the equation as − 2 x = 2 1 x + 4 19 .
Isolate x terms: Subtract 2 1 x from both sides to get − 2 5 x = 4 19 .
Solve for x: Multiply both sides by − 5 2 to find x = 4 19 × − 5 2 .
Simplify: Calculate the value of x as x = − 10 19 .
Explanation
Understanding the Problem We are given the equation − 2 x = − 4 1 + 2 1 x + 5 and we want to find the value of x that satisfies this equation. Our goal is to isolate x on one side of the equation.
Combining Constant Terms First, let's combine the constant terms on the right side of the equation: − 4 1 + 5 = 4 − 1 + 4 20 = 4 19 So the equation becomes: − 2 x = 2 1 x + 4 19
Isolating x Terms Next, we want to get all the x terms on one side of the equation. Let's subtract 2 1 x from both sides: − 2 x − 2 1 x = 4 19
Combining x Terms Now, let's combine the x terms on the left side. To do this, we need a common denominator, which is 2: − 2 4 x − 2 1 x = − 2 5 x So the equation is now: − 2 5 x = 4 19
Solving for x To solve for x , we need to multiply both sides of the equation by the reciprocal of − 2 5 , which is − 5 2 : x = 4 19 × − 5 2
Simplifying the Expression Now, let's simplify the expression to find the value of x : x = − 4 × 5 19 × 2 = − 2 × 2 × 5 19 × 2 = − 2 × 5 19 = − 10 19 So, the solution is x = − 10 19 .
Final Answer Therefore, the value of x that satisfies the given equation is x = − 10 19 .
Examples
In physics, this type of equation could represent a force balance where x is a displacement. Solving for x tells you the equilibrium displacement where the forces balance. For example, imagine a spring attached to a block, with an external force applied. The equation could describe the balance between the spring force and the external force, allowing you to find the block's position at equilibrium. Understanding how to solve such equations is crucial for predicting the behavior of physical systems.
