SOLVING EQUATIONS Using addition property of equality

ADDITION PROPERTY OF EQUALITY

The addition property of equality says that, if you add the same quantity to both sides of an equation, the equality still holds true.

If a = b, then a + c = b + c Think about the equation 3 = 3. It is true. We know that if we were to

add a number to only one side of the equation, the statement would no longer be true. For example, 3 + 1 = 3 is false. But if we added the same amount to both sides of

the equation, it would still be true. Add 1 to both sides: 3 + 1 = 3 + 1; 4 = 4 is true.

Since subtraction is the same as adding the opposite, this property also applies when we wish to subtract the same number from both sides of an equation. For example, 3 + (-1) is the same as 3 - 1

Now, what if the equation is x = 3? We know that 3 is the only value of x that would make the statement true. The result of adding 1 to both sides of the equation x = 3 is:

x + 1 = 3 + 1 or x + 1 = 4Notice that the only value of x that would make this new equation true is still 3.

So, what the addition property of equality says is that we can add the same value to each side of an equation while still maintaining the same solution. A solution is the value of the variable that makes the sentence true.

ADDITION PROPERTY OF EQUALITY CONT.

EXAMPLE Solve x - 4 = 20 for x

Solution: If we solve this example by inspection, we can see the solution is 24. However, we can apply the addition property to get the same result. Add the constant, 4, in order to isolate the variable (get it alone on one side of the equation).

x – 4 = 20Add 4 x – 4 + 4 = 20 + 4

We have x + 0 = 24 And x = 24

Check in the original equation 24 – 4 = 20

ADDITION PROPERTY OF EQUALITY CONT.

EXAMPLE Solve x + 7 = 30 for x.

Solution:x + 7 = 30

Add -7 x + 7 – 7 = 30 - 7x + 0 = 23 x = 23

Check 23 + 7 = 30 30 = 30

So the solution is 23.

ADDITION PROPERTY OF EQUALITY CONT.

EXAMPLE Solve p + (-7) = -12 for p.

Solution:p + (-7) = -12 is the same as p – 7 = -12, so solve

p – 7 = -12Subtract 7 from both sides p – 7 + 7 = -12 + 7

p = -5

Check -5 + (-7) = -12

-12 = -12So the solution is -5

ADDITION PROPERTY OF EQUALITY CONT.

The addition property of equality doesn't just apply to numerical terms. It also allows you to add or subtract algebraic terms (or even entire expressions) from each side of the equation. Since solving an equation requires that the variable is by itself on one side of the equation, it is necessary to get all of the terms containing the variable on one side. Then we can combine the like terms.

ADDITION PROPERTY OF EQUALITY CONT.

EXAMPLE Solve -3 + 4x = 5x

Solution: -3 + 4x = 5x

Subtract 4x from both sides -3 + 4x – 4x = 5x – 4x

Combine like terms -3 = x

Check: -3 + 4(-3) = 5(-3)

-3 + -12 = -15 -15 = -15

So the solution is -3.

ADDITION PROPERTY OF EQUALITY CONT.

EXAMPLE Jan's age in 10 years will be 25, how old is Jan?

Solution:Let J be Jan's age; then we have J + 10 = 25.

Solve J + 10 = 25 using the addition property of equality.

J + 10 = 25 Subtract 10 from both sides J + 10 - 10 = 25 - 10

J = 15Check:If Jan is 15 years old now, then in 10 years, she will be 15 + 10, or 25 years old.