Solve the equation for $x$.
$\frac{2 x-3}{5}=\frac{x}{10}$
Answer (2)
Multiply both sides by 10: 2 ( 2 x − 3 ) = x .
Distribute: 4 x − 6 = x .
Subtract x from both sides: 3 x − 6 = 0 .
Add 6 to both sides and divide by 3: x = 2 .
Explanation
Problem Analysis We are given the equation 5 2 x − 3 = 10 x and asked to solve for x . We will manipulate the equation to isolate x on one side.
Eliminating Fractions First, we multiply both sides of the equation by 10 to eliminate the fractions. This gives us: 10 ⋅ 5 2 x − 3 = 10 ⋅ 10 x Simplifying, we get: 2 ( 2 x − 3 ) = x
Distributing Next, we distribute the 2 on the left side of the equation: 4 x − 6 = x
Isolating x Terms Now, we subtract x from both sides of the equation to get all the x terms on one side: 4 x − x − 6 = x − x 3 x − 6 = 0
Isolating the x Term We add 6 to both sides of the equation to isolate the x term: 3 x = 6
Solving for x Finally, we divide both sides by 3 to solve for x :
x = 3 6 x = 2
Final Answer Therefore, the solution to the equation is x = 2 .
Examples
In real-world scenarios, solving linear equations like this is crucial for determining unknown quantities in various fields such as physics, engineering, and economics. For instance, if you're calculating the distance traveled at a constant speed, or determining the break-even point for a business, you'll often need to solve equations of this form to find the value of a variable.
To solve the equation 5 2 x − 3 = 10 x , we first eliminate the fractions by multiplying both sides by 10. After distributing and isolating the variable, we find that x = 2 .
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