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Solving Multi-Step Linear Equations (page 2 of 3)

Sections: One-step equations, Multi-step equations, "No solution" and "all x" equations

Most linear equations require more than one step for their solution. For instance:

• Solve 7x + 2 = –54

I need to undo the "times seven" and the "plus two". There is no rule about which "undo" I should do first. However, if I first divide through by 7, I'm going to have fractions. Personally, I prefer to avoid fractions if possible, so I almost always do any plus / minus before any times / divide:

Then the solution is x = –8. 

Formatting your homework and showing your work in the manner I have done above is, in my experience, fairly universally acceptable. However (warning!), it is also a good idea to clearly rewrite your final answer at the end of each exercise, as shown (in purple) above. Don't expect your grader to take the time to dig through your work and try to figure out what you probably meant your answer to be. Format your work so as to make your meaning clear!

• Solve –5x – 7 = 108

Then the solution is  x = –23.

• Solve 3x – 9 = 33 Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

Then the solution is  x = 14.

• Solve 5x + 7x = 72

First, I need to combine like terms on the left; then I can solve:

Then the solution is  x = 6.

• Solve 4x – 6 = 6x

I need to move all the x's over to one side or the other. To avoid negative coefficients on my variables, I usually move the smaller x; in this case, I'll subtract the 4x over to the other side:

Then the solution is x = –3.

In the above exercise, note that it is perfectly okay to have the "x=" be on the right. The variable is not "required" to be on the left; we're just used to seeing it there. It's alright if your solution works out with the variable on the right. However (warning!), I have heard of some instructors who insist that the variable be placed on the left-hand side in the final answer. (No, I'm not making that up.) If you have any doubts about your instructor's formatting preferences, ask now.

• Solve 8x – 1 = 23 – 4x

Then the solution is  x = 2.

• Solve 5 + 4x – 7 = 4x – 2 – x

Before I can solve, I need to combine like terms:

Then the solution is x = 0.

It is perfectly fine for x to have a value of zero. Zero is a valid solution. Do not say that this equation has "no solution"; it does indeed have a solution, that solution being x = 0.

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