Combining or Collecting Like Terms

The phrases “combine like terms” or “collect like terms” are used a lot in algebra, and for good reason. The process they describe is used a lot in solving algebra problems. Two approaches, one intuitive and the other algebraic, can help in understanding why some terms are “like” terms, and others are not.

Quantities With Units

Suppose you are sitting in front of a table that holds three piles of fruit:
– five apples
– three oranges
– four apples
If someone asks you “What do you see on the table?”, how would you answer the question?

Chances are you answered “nine apples and three oranges”. Why did you combine the two piles of apples with one another, but not with the oranges? How did you know that you could do that?

The quantities of apples may be combined because addition or subtraction only work with  Continue reading Combining or Collecting Like Terms

Simplifying Fractions

Three concepts help explain the process of simplifying fractions:

1. Multiplying a quantity by 1 has no effect
2. A fraction whose numerator is exactly the same as its denominator is equal to 1 (unless the denominator equals zero)
$\dfrac{17a^2b}{17a^2b}~~=~~1~~~~a\ne 0,~~b\ne 0$
3. A product of two fractions can be rewritten as a fraction of two products (and vice versa)
$\dfrac{a}{b} \cdot \dfrac{c}{d}~~=~~\dfrac{ac}{bd}\\*~\\*\dfrac{ac}{bd}~~=~~\dfrac{a}{b} \cdot \dfrac{c}{d}$

To simplify a fraction:

• Rewrite both numerator and denominator as products of factors (if they are not already factored)
• Examine both numerator and denominator to see if they share any factors
• If they do share factors, use concept (3) above to move the shared factors into a separate fraction
• That separate fraction should now have a numerator that is exactly the same as its denominator, which by concept (2) above means that it must equal 1, therefore by concept (1) above we can drop it from the expression

Consider the following fraction… can it be simplified? Continue reading Simplifying Fractions