Algebra: Equations with Multiple Steps
Equations with Multiple Steps
Most of the time, solving equations requires more than a single step. For example, think about the equation I introduced you to at the beginning of the section: 3x - 2 = 19. Not only is there a -2 on the same side of the equal sign as the x, but there's also a 3 clinging to that x, like a dryer sheet stuck to a pant leg. In order to isolate x (and therefore solve the equation), you'll have to get rid of both numbers, using each of the techniques you've learned so far. (It wouldn't hurt to throw in some fabric softener with static cling controller as well.)
If a solution requires more than a single step, here's the order you should follow:
Talk the Talk
Consider the equation 3y - 7y = 12. Since 3y and -7y both have the exact same variable part (y), they are called like terms and you can simplify by combining the coefficients and leaving the variable alone: 3y - 7y = -4y, since 3 - 7 = -4, so the equation is now -4y = 12.
I'll carefully define like terms and discuss them further in Introducing Polynomials.
- Simplify the sides of the equation separately. Each of the items added to or subtracted from one another in the equation are called terms. If two terms have the exact same variable portion, then they are called like terms, and you can combine them as though they were numbers.
- Separate the variable. Using addition and subtraction, move all terms containing the variable you're isolating to one side of the equation (usually the left) and move everything else to the other side (usually the right). You're finished when you have something that looks like this: ax = b (a number times the variable is equal to a number).
- Eliminate the coefficient. If the variable's coefficient is something other than 1, you need to either divide by it or multiply by its reciprocal (like you did earlier in this section).
Equation solving requires practice, and it's going to take some trial and error before you get good at it. Don't forget to check your answers! Even though I will only show answer checking rarely from this point forward (to save space), rest assured that I never let the chance to make sure I got the answer right go by! Eventually, you'll feel comfortable checking answers by substituting in your head and working things out mentally.
Example 2: Solve each equation.
- (a) 3x - 2 = 19
- Solution: You can't simplify the left side, since 3x and -2 are not like terms, so the first thing to do is to separate the variable term. Accomplish this by adding 2 to both sides.
Divide both sides by 3 to eliminate the coefficient.
- 3x⁄3 = 21⁄3
- x = 7
- (b) -14 = 2x + 4(x + 1)
- Solution: You can do a bit of simplifying on the right side of the equation. Start by distributing that positive 4 into the quantity within parentheses.
- -14 = 2x + 4 · x + 4 · 1
- -14 = 2x + 4x + 4
- Simplify like terms 2x and 4x.
- -14 = 6x + 4
- At this point, the problem looks a lot like the equation from part (a), except the variable term appears on the right side of the equation. There's no problem with that—it's perfectly fine. In fact, if you leave the 6x on the right side, it's less work to separate the variable term. Just subtract 4 from both sides.
Divide both sides by 6 to eliminate the coefficient.
- -18⁄6 = 6x⁄6
- -3 = x
How'd You Do That?
In Example 2, part (b), I solved the equation by isolating the x on the right side, rather than the left side. To tell you the truth, I prefer x on the left side as a matter of personal taste, even though it doesn't affect the answer at all.
According to the symmetric property of algebra, you can swap sides of an equation without affecting its solution or outcome. In other words, I could have flip-flopped the sides of the equation in 2(b) to get 2x + 4(x + 1) = -14. If you solve that equation, you'll get x = -3, the exact same answer. So, if you ever wish the sides of an equation were reversed, go ahead and flip them without fear.
- (c) -3(x + 7) = -2(x - 1) + 5
- Solution: You can apply the distributive property on both sides of the equal sign to begin.
- -3x - 21 = -2x + 2 + 5
- Simplify the right side by combining the 2 and 5 (which are technically like terms, since they have the exact same variable part—no variables at all).
- -3x - 21 = -2x + 7
- Now, it's time to separate the variable term. Do this by adding 2x to both sides (to remove all x terms from the right side of the equation) and adding 21 to both sides as well (to remove plain old numbers from the left side of the equation).
As demonstrated in Example 2(c), a negative variable like -w technically has an implied coefficient of -1, so you can rewrite it as -1w if you wish. (This is similar to implied exponents, where a plain old variable like w has an implied coefficient of 1, so w = 1w1.)
- At this point, you have -x = 28, which means "the opposite of the answer equals 28." Therefore, the correct answer is x = -28 (since -28 is the opposite of 28).
- Here's another way to get the final answer: Since -x = -1 · x, you can rewrite the final line of the equation so it looks like it has a coefficient and divide by that -1 coefficient:
- -1x⁄1 = 28⁄1
- x = -28
- (d) y + 3 = 1⁄4y + 5
- Solution: Since there are no like terms together on one side of the equation, skip right to separating the variable terms. Accomplish this by subtracting 1⁄4y and 3 on both sides of the equation. (By the way, even though the variable is y, not x, in this equation, that doesn't change the way you solve it.)
You've Got Problems
Problem 3: Solve the equations.
(a) 3(2x - 1) = 14
(b) 2x - 7 = 4x + 13
- Since the coefficient is fractional, multiply both sides by its reciprocal to finish.
- 4⁄3 (3⁄4y = (4⁄3) 2⁄1
- 12⁄12y = 8⁄3
- y 8⁄3
- Since 8 and 3 have no common factors (other than 1), the improper fraction cannot be simplified, so leave it as is.
Excerpted from The Complete Idiot's Guide to Algebra © 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.