06.08.2018: One Chapter of Math

Today’s soundtrack is Noah Gundersen: Carry the Ghost.

This morning, I’m working on chapter 22 of Basic Math & Pre-Algebra For Dummies, “Unmasking Mr. X: Algebraic Equations,” in which I will learn how to find out what number a variable stands for, which is “called solving for x” (p. 299).

There are four ways to solve algebraic equations: “Eyeballing easy equations[, …] Rearranging slightly harder equations[, …] Guessing and checking equations[, and] Applying algebra to more-difficult equations” (p. 302). Examples given are as follows:

  • 1+x=4 - we can eyeball this and easily see that x = 3.

  • 5x = 65 - we can rearrange this to find that 65 / 5 = 13; therefore, x = 13.

  • 2x + 14 = 20 - we can guess and check this one. If we solve as x = 4, does it work? No. But x = 3 works! 2 x 3 = 6, and 6 + 14 = 20.

  • “11x - 13 - 9x + 3” (p. 302) - I can’t easily use the previous three methods to solve this one, so I’ll need to use algebra to solve it. To do so, I’ll need to use the “balance scale method” (p. 302).

To perform a balanced scale equation, we must remember the fundamental principle that “you can add any number to [an] equation, as long as you do it to both sides[, and] you can [...] subtract an x, or even multiply or divide by x, as long as you do so to both sides” (p. 303).

The goal when solving for x is to “isolate x -- that is, to get x alone on one side of an equation and some number on the other side” (p. 304), which in lower-difficulty algebra involves three steps:

“1. Get all constants (non-x terms) on one side of the equation.

2. Get all x-terms on the other side of the equation.

3. Divide to isolate x” (p. 304).

When solving more-difficult equations, we can also rearrange equations as needed. If parentheses are involved, we can do so with the FOIL method that was covered in chapter 21. If fractions are involved, we can use cross-multiplication. To do so, we multiply the top-left by the bottom-right, and say that it equals the bottom-left multiplied by the top-right. We simplify if needed.