Algebraic steps to simplify an equation

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In summary, the conversation involves a student asking for help with algebraic steps involved in solving an equivalency problem. Another user provides guidance on how to simplify and factor the given expression. The student eventually figures out the solution and thanks the other user for their help.
  • #1
JOhnJDC
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Homework Statement


Could someone show me the algebraic steps involved in the following equivalency?

Homework Equations



img8.gif


The Attempt at a Solution



EDIT: see attempt below. I eventually answered my own question. Thanks, Mark44.
 
Last edited:
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  • #2
JOhnJDC said:

Homework Statement


Could someone show me the algebraic steps involved in the following equivalency?

Homework Equations



img8.gif


The Attempt at a Solution



I get lost in the algebra every time.

[tex]2(q + \frac{pq}{a})(\frac{-rq}{a^2}) + 2(p + a)[/tex]
[tex]=2(\frac{-pq^2}{a^2} + \frac{-p^2q^2}{a^3}) + 2(p + a)[/tex]

Get a common denominator for the two terms in the first pair of parentheses, and then combine those two terms.
Can you carry on from there?
 
  • #3
Mark44 said:
[tex]2(q + \frac{pq}{a})(\frac{-rq}{a^2}) + 2(p + a)[/tex]
[tex]=2(\frac{-pq^2}{a^2} + \frac{-p^2q^2}{a^3}) + 2(p + a)[/tex]

Get a common denominator for the two terms in the first pair of parentheses, and then combine those two terms.
Can you carry on from there?

Okay, the two terms in parentheses now have a common denominator:

2(a+p) + 2(-apq^2 - p^2q^2)/a^3

This is where I get stuck. I don't know how to factor 2(a+p) out of the numerator 2(-apq^2 - p^2q^2).

Here is my attempt:

2a+2p + (-2apq^2 - 2p^2q^2)/a^3

(2a+2p)[1 + (-pq^2 - pq^2)/a^3]

2(a+p)[1 + (-2pq^2)/a^3]

What am I doing wrong? Why do I end up with -2pq^2/a^3 instead of just pq^2/a^3? Thanks.

EDIT: I got it. I was thinking about the operation in the wrong way. Instead of trying to factor out (a+p), I decided to factor out -pq^2 from 2(-apq^2-p^2q^2)/a^3, which yielded 2[-pq^2(a+p)/a^3]. From there, I could easily factor out 2(a+p) from 2(a+p) + 2(-apq^2 - p^2q^2)/a^3.

Thanks.
 
Last edited:

1. How do I simplify an algebraic equation?

Simplifying an algebraic equation involves using algebraic steps to combine like terms, eliminate parentheses, and apply the rules of exponents. It is important to start by identifying like terms and combining them, then distributing any coefficients, and finally simplifying any remaining terms.

2. What are the basic algebraic steps for simplifying an equation?

The basic algebraic steps for simplifying an equation include combining like terms, distributing coefficients, and applying the rules of exponents. These steps can be repeated multiple times until the equation is simplified as much as possible.

3. Why is it important to simplify an equation?

Simplifying an equation helps to make it easier to understand and work with. It also allows for easier substitution and solving for variables. In addition, simplifying an equation can help to identify any mistakes or errors in the original equation.

4. How do I know if an equation is fully simplified?

An equation is considered fully simplified when there are no more like terms that can be combined, no more parentheses that can be eliminated, and all exponents are applied correctly. This means that the equation cannot be simplified any further using algebraic steps.

5. Are there any common mistakes to avoid when simplifying an equation?

Yes, some common mistakes to avoid when simplifying an equation include forgetting to distribute a coefficient, combining unlike terms, and making errors with the rules of exponents. It is important to carefully check each step and double check the final simplified equation for accuracy.

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