1. Combining Like Terms to Solve Equations
Tutorial 6c

2. Definitions
An equation is created when two mathematical expressions, that are made up of terms, are set equal to one another.
3x – 12 = 7x + 2
A term is a number or a variable, or the product or quotient of numbers and variables.
3x, 12, 7x, and 2 are all terms in the above equation
If a term has a variable, the numerical factor is called the coefficient.
3 and 7 are coefficients

3. Identifying Coefficients
To identify the coefficients of each term in the expression –x + 7y + 4, you can write the expression as –1x + 7y + 4.
3 terms
3 terms
3 terms
3 terms
3 terms
3 terms
3 terms
Coefficients –1 7
–1x +7y + 4
Constant 4

4. Like Terms Terms are like terms if they meet two criteria:
The terms must have the same variable(s).
The variable(s) must have the same exponents on them.
4x and 11x2 are not like terms. They do not have the same exponent.
5x and 7x are like terms. They meet both criteria above.

5. The simplified expression is 8x + 6.
Example #1
You can combine like terms by adding the coefficients.
Simplify: 3x x – 2
3x + 5x – 2  Then add or subtract.
8x 6
The simplified expression is 8x + 6.
Terms with x Constant terms  Group like terms

6. The simplified expression is 4x – 3z + 6.
Example #2
Simplify: x + z – 2x + 8 – 4z
Terms with x Terms with z Constant terms  Group like terms
6x - 2x z – 4z  Then add or subtract.
4x z 6
The simplified expression is 4x – 3z + 6.

7. Solving Equations Solve and then check your answer: 7x – 3x – 6 = 6
Combine like terms.
4x – 6 = 6
To undo the subtraction, add a 6 to each side of the equation. 1 3
4x = 12
To undo the multiplication, divide by 4 on each side of the equation. 1 1
x is now all alone on one side of the equation!
x = 3
Always check your answers!
7x – 3x – 6 = 6; Does x =3?
7•3 – 3•3 – 6 = 6 is true!

8. Problem Solving
As a gift for your brother, you order concert tickets. Two days before the concert a friend asks to join you. When you order the third ticket, there is a $12 charge for express delivery to be sure the ticket arrives on time. The total bill is $78. How much does each person (yourself and your friend) owe?
Define your variable: Let c = charge for each original ticket.
Let c + 12 = total charge for the late ticket.
Relate:
Charge for the
original 2 tickets
Total charge for
the late ticket
plus
is 78
Write: 2c +
C + 12 = 78
Solve: 2c + c + 12 = 78

9. Always check your answers!
Let c = charge for each original ticket.
Let c + 12 = total charge for the late ticket.
Problem Solving
As a gift for your brother, you order concert tickets. Two days before the concert a friend asks to join you. When you order the third ticket, there is a $12 charge for express delivery to be sure the ticket arrives on time. The total bill is $78. How much does each person (yourself and your friend) owe?
2c + c = 78
Combine like terms.
3c + 12 = 78
To undo the addition, subtract a 12 from each side of the equation. 1 22
3c = 66
To undo the multiplication, divide by 3 on each side of the equation. 1 1
You owe 2•($22) = $44
Your friend owes $22 + $12 = $34
c = 22
Always check your answers!
$44 + $34 = $78 

10. Distributive Property
For all real numbers a, b, c:
a(b + c) = ab + ac (b + c)a = ba + ca
a(b – c) = ab – ac (b – c)a = ba – ca
Examples:
3(2x + 5) = 3(2x) + 3(5) = 6x + 15
(5b – 6)4 = (5b)4 – (6)4 = 20b – 24

11. Solving Equations Solve and then check your answer:
6x + 12(10 – x) = 84
6x + 12 (10 – x) = 84
6x + 12•(10) – 12x = 84
Distribute.
6x – 12x = 84
Combine like terms.
-6x = 84
To undo the addition, subtract a 120 to each side of the equation. 1
-6x = -36 6
To undo the multiplication, divide by -6 on each side of the equation. 1 1
x = 6
Always check your answers!
6x + 12(10 – x) = 84; Does x =6?
6•6 + 12(10 – 6) = 84 is true!

12. The End
Time to move on
to the assignment or
the next lesson