rewrite the equation in terms of u.

3 min read 09-02-2025

Rewriting equations in terms of a specific variable, like "u," is a fundamental algebraic skill. This process, often called "solving for a variable," involves manipulating the equation using algebraic operations until the desired variable is isolated on one side of the equals sign. This guide will walk you through various techniques and examples to master this essential concept.

Understanding the Goal

Before diving into the methods, let's clarify the objective. When we say "rewrite the equation in terms of u," we mean to isolate the variable 'u' on one side of the equation, expressing it as a function of other variables in the equation. All other variables will be on the other side.

Methods for Rewriting Equations in Terms of U

The specific steps involved depend on the complexity of the equation. However, the core principles remain consistent: use inverse operations to undo the operations performed on 'u'.

1. Linear Equations

Linear equations involve only the first power of the variable. Solving these is relatively straightforward.

Example: Solve for 'u' in the equation 3u + 5 = 14

Subtract 5 from both sides: 3u = 9
Divide both sides by 3: u = 3

Therefore, the equation rewritten in terms of u is simply u = 3.

2. Quadratic Equations

Quadratic equations contain the variable raised to the power of 2 (u²). These require slightly more advanced techniques.

Example: Solve for 'u' in the equation u² + 6u + 8 = 0

This equation can be solved using factoring, the quadratic formula, or completing the square. Let's use factoring:

Factor the quadratic expression: (u + 2)(u + 4) = 0
Set each factor to zero and solve:
- u + 2 = 0 => u = -2
- u + 4 = 0 => u = -4

In this case, the equation rewritten in terms of u gives us two solutions: u = -2 and u = -4.

3. Equations with U in Multiple Terms

When 'u' appears in multiple terms, you need to combine like terms before isolating 'u'.

Example: Solve for 'u' in the equation 2u + 5u - 7 = 18

Combine like terms: 7u - 7 = 18
Add 7 to both sides: 7u = 25
Divide both sides by 7: u = 25/7

Thus, the equation rewritten in terms of u is u = 25/7.

4. Equations Involving Fractions

Equations with fractions require a little extra care. Often, the first step is to eliminate the fractions by multiplying both sides by the least common denominator (LCD).

Example: Solve for 'u' in the equation u/2 + 3 = 7

Subtract 3 from both sides: u/2 = 4
Multiply both sides by 2: u = 8

The equation rewritten in terms of u is u = 8.

5. Equations with Radicals (Square Roots)

If 'u' is under a square root, you must square both sides of the equation to eliminate the radical. Remember to check your solutions to avoid extraneous solutions (solutions that don't satisfy the original equation).

Example: Solve for 'u' in the equation √u + 2 = 5

Subtract 2 from both sides: √u = 3
Square both sides: u = 9

Check: √9 + 2 = 5 (This is true, so u = 9 is a valid solution). The equation rewritten in terms of u is u = 9.

6. Equations with Exponents

Solving for 'u' in equations with exponents may require using logarithms or other exponential rules.

Example: Solve for 'u' in the equation 2u = 16

Rewrite 16 as a power of 2: 2u = 24
Since the bases are equal, the exponents must be equal: u = 4

The equation rewritten in terms of u is u = 4.

Practical Applications

Rewriting equations in terms of a specific variable is crucial in many fields:

Physics: Solving for unknown variables in physics formulas.
Engineering: Calculating unknown parameters in design equations.
Economics: Modeling economic relationships and predicting outcomes.
Computer Science: Developing algorithms and solving computational problems.

Mastering this skill will significantly enhance your ability to solve problems in various disciplines. Remember to practice regularly with diverse equation types to build your proficiency. The more you practice, the easier it will become to confidently rewrite equations in terms of any variable.