Math Problem Statement

Solve the equation: 1/2x + 3 - 1/4x = 3 + 1/4x

Solution

The equation in the image is:

12x+314x=3+14x\frac{1}{2}x + 3 - \frac{1}{4}x = 3 + \frac{1}{4}x

Let's solve this step-by-step:

Step 1: Simplify the equation

Move all the terms involving xx to one side and constants to the other side.

12x14x14x=33\frac{1}{2}x - \frac{1}{4}x - \frac{1}{4}x = 3 - 3

Step 2: Combine like terms

The left side has xx-terms and the right side has constants. First, simplify the xx-terms.

(12x14x14x)=0\left( \frac{1}{2}x - \frac{1}{4}x - \frac{1}{4}x \right) = 0

Combine the fractions:

12x2(14x)=12x24x=12x12x=0\frac{1}{2}x - 2\left(\frac{1}{4}x\right) = \frac{1}{2}x - \frac{2}{4}x = \frac{1}{2}x - \frac{1}{2}x = 0

This gives:

0=00 = 0

Step 3: Interpret the result

Since both sides of the equation simplify to 0=00 = 0, this indicates that the equation is true for all values of xx. Therefore, the solution is that any real number xx is a solution.

Would you like further details or have any questions?


Here are 5 related questions to explore:

  1. What are the possible types of solutions in linear equations (one solution, no solution, or infinitely many solutions)?
  2. How do we handle equations involving fractions with different denominators?
  3. What is the geometric interpretation of equations that are true for all values of xx?
  4. How would the solution change if the equation had terms with x2x^2 or other powers?
  5. What strategies can be used to simplify equations with fractions?

Tip: When dealing with fractions in equations, it's often useful to clear them by multiplying through by the least common denominator (LCD).

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Fractions

Formulas

Simplification of linear equations
Combining like terms

Theorems

Linear equation theorem
Zero property of equality

Suitable Grade Level

Grades 8-10