Combining Like Termsto 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.

–1x +7y + 4

3 terms3 terms3 terms3 terms3 terms3 terms3 terms

Coefficients

–1 7

Constant

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.

Example #1

You can combine like terms by adding the coefficients. Simplify: 3x + 8 + 5x – 2

3x + 5x 8 – 2 Then add or subtract. 8x 6

The simplified expression is 8x + 6.

Terms with x Constant terms Group like terms

Example #2

Simplify: -2 + 6x + z – 2x + 8 – 4z

Terms with x Terms with z Constant terms Group like terms

6x - 2x 1z – 4z -2 + 8 Then add or subtract. 4x -3z 6

The simplified expression is 4x – 3z + 6.

Solving EquationsSolve and then check your answer: 7x – 3x – 6 = 6

To undo the subtraction, add a 6 to each side of the equation.

Always check your answers! 7x – 3x – 6 = 6; Does x =3?7•3 – 3•3 – 6 = 6 is true!

4x – 6 = 6

x = 3

4 4

1 3

1 1

4x = 12 To undo the multiplication, divide by 4 on each side of the equation.

x is now all alone on one side of the equation!

Combine like terms. 7x – 3x – 6 = 6

+ 6 + 6

Problem SolvingAs 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 theoriginal 2 tickets

plus Total charge forthe late ticket

is 78

Write: 2c + C + 12 = 78

Solve: 2c + c + 12 = 78

Problem SolvingAs 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?

To undo the addition, subtract a 12 from each side of the equation.

Always check your answers!$44 + $34 = $78

3c + 12 = 78

c = 22

3 3

1 22

1 1

3c = 66 To undo the multiplication, divide by 3 on each side of the equation.

Combine like terms. 2c + c + 12 = 78

- 12 -12

You owe 2•($22) = $44Your friend owes $22 + $12 = $34

Let c = charge for each original ticket.Let c + 12 = total charge for the late ticket.

Distributive Property

For all real numbers a, b, c:a(b + c) = ab + ac (b + c)a = ba + caa(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 EquationsSolve and then check your answer: 6x + 12(10 – x) = 84

To undo the addition, subtract a 120 to each side of the equation.

Always check your answers! 6x + 12(10 – x) = 84; Does x =6?6•6 + 12(10 – 6) = 84 is true!

-6x + 120 = 84

x = 6-6 -6

1 6

1 1

-6x = -36 To undo the multiplication, divide by -6 on each side of the equation.

Combine like terms.

6x + 12 (10 – x) = 84

-120 -120

Distribute. 6x + 12•(10) – 12x = 84

6x + 120 – 12x = 84