Elimination using addition and subtraction

Elimination using addition and subtraction means solving an algebraic equation by finding the value of the unknown. The goal is to always isolate the variable—this means to get the variable all by itself on one side of the equation while the rest of the equation to the other side. This can usually be accomplished by performing inverse operations that result in an equivalent equation that has the same solution as the original equation.

Consider the algebraic equation,

x + 3 = 4

Likely, without having to resort to any written calculations, doing mental math you know that if x is 1 (or when x = 1), the above equation is satisfied, since 1 + 3 = 4. However, doing problems that involve more than one step or operation to solve for the unknown variable becomes complex to do orally.

To prepare for these multiple-step problems, you need to consider the value of x more formally or use elimination using addition and subtraction. Our goal is always to isolate the variable x; however, observe that the constant 3 has been added to the variable. You can isolate the variable x by subtracting 3 from both sides of the equation (which is the opposite or inverse of adding). Thus, as a result of doing the opposite operation (or inverse operation), we have

x + 3 – 3 = 4 – 3

Now on the left side of the equation, 3 – 3 = 0, only the variable x remains―which is to be expected since you did an inverse operation; on the right side of the equation, we evaluate 3 – 2 which is 1. Thus, after simplifying both sides of the equation we have a solution,

x = 1

In general, if you have an algebraic equation of the form x + a = b, where a and b are constants, you can use the subtraction property of equality to form the new equivalent equation x + a – a = b – a. This effectively isolates the variable x, yielding the desired solution x = b – a.

Elimination using addition and subtraction example:

Consider another algebraic equation—this time the variable is on the right side of the equation:

36 = x + 18

Again, the goal here is to isolate the variable x. you need to perform the inverse operation by subtracting 18 from each side. Showing the details of this step, the equivalent equation become

36 – 18 = x + 18 – 18

Simplifying the equation, we have 36 – 18 = x + 0

17 = x

It is more convenient to write x = 17 as the solution (with the variable written first and set equal to the value that makes the initial equation true). We have written x = 17 instead of 17 = x, this doesn’t change the value of the equation. You can always interchange the left-hand side and right-hand side of the equation if needed. In the above example, you could have considered the equation as x + 18 = 36 (instead of 36 = x + 18) and the solution would have again been x = 17.

In general, the fact that any equation of the form a = b can be written as b = a is called the symmetric property of equality.

When you subtract, add, multiply or divide the same number from both sides of the equation, you are using the subtraction, addition, multiplication, and division property of equality respectively.

Now, consider another equation

x – 4 = 6

You observe that the constant 4 is subtracted from x on the left side of the equation. To solve for x, you must do the inverse operation by adding 4 to both sides of the equation, or

x – 4 + 4 = 6 + 4

After doing the inverse operation and simplifying, we have

x = 10

For linear equation in two variables click here

Elimination using addition and subtraction: Examples – Solve for the unknown variable in each equation.

Question 1: x – 7 = 10

x – 7 = 10

x – 7 + 7 = 10 + 7

x = 17

Question 2: x + 7 = 10

x + 7 = 10

x + 7 – 7 = 10 – 7

x = 3

Question 3: x – 7 = – 10

x – 7 = -10

x – 7 + 7 = -10 + 7

x = -3

Question 4: x + 7 = – 10

x + 7 = -10

x + 7 – 7 = -10 – 7

x = -17

Question 5: -4 = x + 3

-4 = x + 3

x + 3 = -4

x + 3 – 3 = -4 -3

x = -7

Question 6: -4 = x – 3

-4 = x – 3

x – 3 = -4

x – 3 + 3 = -4 + 3

x = -1

Question 7: x +5 = 17

x + 5 = 17

x + 5 – 5 = 17 – 5

x = 12

Question 8: x + 9 = 17

x + 9 = 17

x + 9 – 9 = 17 – 9

x = 8

Conclusion:

if you have an algebraic equation of the form x – a = b, where a and b are constants, you can use the addition property of equality to form the new equivalent equation

x – a + a = b + a.

x = b + a