Transcript Combining Like Terms to Solve Equations

Combining Like Terms
to Solve Equations
Tutorial 6c
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
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
–1x +7y
–1
7 +4
Coefficients
Constant
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.
Example #1
You can combine like terms by adding the
coefficients.

Simplify:
Terms with x
3x + 5x
8x
3x + 8 + 5x – 2
Constant terms
8–2
6
 Group like terms
 Then add or subtract.
The simplified expression is 8x + 6.
Example #2
Simplify:
Terms with x
Terms with z
6x - 2x
4x
1z – 4z
-3z
-2 + 6x + z – 2x + 8 – 4z
Constant terms  Group like terms
-2 + 8
6
 Then add or subtract.
The simplified expression is 4x – 3z + 6.
Solving Equations
Solve and then check your answer:

7x – 3x – 6 = 6
 Combine like terms.
 To undo the subtraction, add a 6
to each side of the equation.
 To undo the multiplication, divide
by 4 on each side of the equation.
 x is now all alone on one side of
7x – 3x – 6 = 6
4x – 6 = 6
+6 +6
14x = 12 3
1
the equation!
Always check your answers!
7x – 3x – 6 = 6; Does x =3?
7•3 – 3•3 – 6 = 6 is true!
4
4 1
x = 3
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
plus
Total charge for
the late ticket
is
78
+
C + 12
=
78
Write:
2c
Solve:
2c + c + 12 = 78
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 + 12 = 78
 Combine like terms.
 To undo the addition, subtract a
12 from each side of the equation.
3c + 12 = 78
- 12 -12
 To undo the multiplication, divide
13c = 66 22
You owe 2•($22) = $44
Your friend owes $22 + $12 = $34
1
by 3 on each side of the equation.
Always check your answers!
$44 + $34 = $78 
3
31
c = 22
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
Solving Equations
Solve and then check your answer:

6x + 12(10 – x) = 84
 Distribute.
 Combine like terms.
 To undo the addition, subtract a
120 to each side of the equation.
 To undo the multiplication, divide by
-6 on each side of the equation.
6x + 12 (10 – x) = 84
6x + 12•(10) – 12x = 84
6x + 120 – 12x = 84
-6x + 120 = 84
-120 -120
1-6x = -36 6
-6
-6 1
1
x = 6
Always check your answers!
6x + 12(10 – x) = 84; Does x =6?
6•6 + 12(10 – 6) = 84 is true!