Lesson #4 Solving Linear Equations

Our third basic Algebra task is found in the instruction "Solve". To solve an equation means to find the value that the variable must have so that the equation is a true statement. The basic "solve" problem consists of an equation with one variable. An equation is any two variable expressions with an "=" sign between them. Think of the two sides of the equal sign separately. These are two different expressions (you may be able to simplify each side on its own). The equal sign provides the information that whatever the two sides look like, they represent the exact same value. For example, when we say

Solve 4x - 12 = 2 + 3(x - 4),

we mean that, for some value of x, "4x - 12" and "2 + 3(x - 4)" have exactly the same value. Although the two expressions may have different values, depending on the value of x, our task is to find the one value of x that makes the two sides equal. This correct value for x is called the solution of the equation.

We can verify that a given value is a solution to an equation much like the way we evaluate expressions: Simply replace the variable with the given value and see whether it makes the equation true.

Example 1. Check that x = 2 is a solution to the equation 4x - 12 = 2 + 3(x - 4).

Replace each occurrence of the letter x with the value 2 to get:

4(2) - 12 = 2 + 3(2 - 4)

8 - 12 = 2 + 3(-2)

-4 = 2 - 6

-4 = -4

Since the resulting values of each side are the same, we can conclude that is indeed a solution to the equation.

Example 2. Is y = -5 a solution to 2y + 3 = y + 12?

Replace y with (-5) on both sides:

2(-5) + 3 = (-5) + 12

-10 + 3 = -5 + 12

-7 = +7

This is a false statement (-7 does not equal +7). So y = -5 is not a solution.

 

Exercises

Test whether a given value is a solution by substitution.

1) Is x = 1 a solution to x + 7 = 3x + 5?

2) Is x = -3 a solution to 4 + 2x = 2(x + 1)?

 

 

3) Is t = 2 a solution to ?

4) Is a = -1 a solution to 3(a + 1) = a + 2?

 

Next we turn to the task of solving the equations for ourselves. How do we find the solution when we are only given the equation? This is perhaps the most important task in Algebra. We will follow one simple guiding principle: Every change you make to simplify an equation must be balanced. That is, to preserve the truth of the equation, you must do the same thing to both sides at each step. In particular, you may do any of the following with an equation:

  • Simplify one side by replacing it with a simpler, equivalent expression (this is just like "Simplifying");
  • Add the same number to both sides of the equation;
  • Subtract the same number from both sides;
  • Multiply both sides by the same number;
  • Divide both sides by the same number.

 

In short, you can apply any operation to an equation, as long as you do it to both sides. Remember that the equation is telling you that these two expressions are equal. To keep them equal, you must treat them the same way.

Example 3. Solve the equation: x + 9 = 14.

The goal is to isolate the variable x by moving any extra numbers and operations. This equation currently reads "a number plus 9 equals 14." You have likely already solved this in your own mind by subtracting 9 from 14. This is the same work we want to show algebraically as follows:

x + 9 =

14

 

-9

-9

We subtract 9 from both sides of the equation.

x =

5

The resulting equation shows x (now by itself) is 5.

We chose to "subtract 9" from both sides because this operation exactly canceled the "plus 9" that started on the side with the variable. Always use the opposite operation to move a number away from the variable.

Example 4. Solve the equation: x - 12 = 8.

This time we want to move the "- 12" away from the variable.

x - 12 =

8

 

+12

+12

We add 12 to both sides of the equation.

x =

20

The resulting equation shows x is 20.

More Exercises

Solve each of the following equations by adding or subtracting one number from each side:

5) x - 3 = 5

6) x + 4 = 6

7) y + 5 = 2

8) x - 10 = 12

9) a + 6 = 4

10) t - 7 = -4

11)

12)

13) 2 + x = 9

14) -3 + m = -8

15) x + 412 = 591

16) y - 92 = 63

Example 5. Solve the equation: 3x = 18.

This equation reads "3 times a number is 18." Note that the operation involved now is multiplication. To move the "3 times" we will divide by 3 since division is opposite to multiplication.

Note that the 3's cancel on the left side, leaving only the x. Divide on the right side to get the final result (the answer may be left as a fraction).

 

More Exercises

Solve each of the following equations by dividing both sides by the same number (Improper fractions are okay for an answer):

17) 2x = 10

18) 5t = 20

19) 3x = -21

20) 9x = -18

21) 10y = 90

22) 7a = 10

23) 2x = 15

24) 6y = -14

25) 2x = 12

26) -8y = 20

27) -6x = -9

28) 5x = 25

29) 12c = -18

30) -4x = -12

31) -2x = 13

32) 20t = -16

 

 

 

 

Example 6. Solve the equation: 5x - 14 = -4.

All of the earlier exercises and examples could be solved in only one step since there was only one number to be moved. Start by adding 14 to both sides so that the variable term (5x) is then alone. Then divide both sides by 5.

5x - 14 = -4

+14 +14

5x = 10

x = 2

 

More Exercises

Solve each of the following equations in two steps (first add/subtract a number, then divide a number):

33) 2x - 3 = 5

34) 3y + 4 = -11

35) -5m + 6 = 13

36) -2x + 1 = -3

37) 9y - 8 = 14

38) 6a + 2 = 19

39) -3x - 4 = -5

40) -8t + 3 = 15

41) 5 + 6y = 12

42) 2 - 3x = 10

43) 7c + 1 = 15

44) 9 - 4x = 6

45) r - 3 = 3

46) 5w + 210 = 600

47) 4y + 7 = 7

48) 22 - 20x = 18

 

Solve each of the following equations by simplifying one side first.

49) 3(x - 2) = 8

50) 2(4y + 1) = 10

51) 6(3 - 2x) = 15

52) 6 + 2(3x - 1) = 4

53) 3 + 4(5x - 8) = 21

54) 10 - 2(6x + 9) = -4

 

Check your answers for Lesson #4.

Go on to Lesson #5.

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