Answer:
4. x=4
5. x=-8
6. x=8
Explanation:
It may take one extra step to get to the solution, but this method always works.
1. find the variable term that is smallest or most negative. Subtract all the terms on that side of the equation from both sides of the equation.
2. collect terms
3. divide the equation by the coefficient of the variable
4. add the opposite of the constant
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4. The most negative variable term is -9x, which is on the left side. Subtracting (24-9x) from both sides of the equation, we have ...
0 = -3x -24 +9x
0 = -24 +6x
0 = -4 +x . . . . . divide by 6
4 = x . . . . . . . . add the opposite of -4
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5. The smallest variable term is 18x, on the right. (The variable term on the left is 20x.)
4(5x +2) +11 -18x -3 = 0 . . . subtract the right side
2x +16 = 0 . . . collect terms
x +8 = 0 . . . . . . divide by 2
x = -8 . . . . . . . . add -8
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6. All variables are on the left side, so we can just collect terms and divide by the coefficient of the variable.
-5x = -40 . . . collect terms
x = 8 . . . . . divide by -5
If you were to literally follow the steps above, you would recognize that -5x is less than 0x (the x-term on the right side of the equation), so you would subtract the left side, giving ...
0 = 5x -40
0 = x -8 . . . . . divide by 5
8 = x . . . . . . . . add 8
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Comment on this solution technique
You will often be told to solve these equations by separating the variable terms from the constant terms. This method actually puts the variable terms and constant terms together (and zero on the other side of the equal sign). The constant is separated from the variable as the last step of this solution process, rather than as one of the first steps. By doing this, we don't have to worry about which variable term or which constant term we're going to mess with.
The only reason for choosing the variable term with the smallest (least) coefficient in the first step is to ensure that the resulting variable coefficient is positive. This tends to reduce errors later on. You can also use that same strategy when solving the equation following the "separate constant terms and variable terms" approach.