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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
2.6
Solving Equations: The Addition and
Multiplication Properties
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 22
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Statements like 5 + 2 = 7 are called equations.
An equation is of the form expression = expression.
An equation can be labeled as
Equal sign
left side right side
x + 5 = 9
Equations
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Solutions of Equations
When an equation contains a variable, deciding which values of the variable make an equation a true statement is called solving an equation for the variable.
A solution of an equation is a value for the variable that makes an equation a true statement.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Solutions of Equations
Determine whether a number is a solution:
Is – 2 a solution of the equation 2y + 1 = – 3?
Replace y with -2 in the equation.
2y + 1 = – 3
2(– 2) + 1 = – 3?
– 4 + 1 = – 3
– 3 = – 3
?
TrueTrue
Since – 3 = – 3 is a true statement, – 2 is a solution of the equation.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Solutions of Equations
Determine whether a number is a solution:
Is 6 a solution of the equation 5x – 1 = 30?
Replace x with 6 in the equation.5x – 1 = 30
5(6) – 1 = 30?
30 – 1 = 30
29 = 30
?
FalseFalse
Since 29 = 30 is a false statement, 6 is not a solution of the equation.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Solving Equations
To solve an equation, we use properties of equality to write simpler equations, all equivalent to the original equation, until the final equation has the form
x = number or number = x
Equivalent equations have the same solution.
The word “number” above represents the solution of the original equation.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Addition Property of Equality
Let a, b, and c represent numbers.
If a = b, then
a + c = b + c
and
a – c = b c
In other words, the same number may be added to or subtracted from both sides of an equation without changing the solution of the equation.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 88
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Solve for x.
x 4 = 3To solve the equation for x, we need to rewrite the
equation in the form x = number. To do so, we add 4 to both sides of the equation. x 4 = 3 x 4 + 4 = 3 + 4 Add 4 to both sides. x = 7 Simplify.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 99
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Check
x 4 = 3 Original equation
7 4 = 3 Replace x with 7.
3 = 3 True.
Since 3 = 3 is a true statement, 7 is the solution of the equation.
To check, replace x with 7 in the original equation.
?
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1010
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Note that it is always a good idea to check the solution in the original equation to see that it makes the equation a true statement.
Helpful Hint
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1111
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Remember that we can get the variable alone on either side of the equation. For example, the equations
x = 3 and 3 = xboth have a solution of 3.
Helpful Hint
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1212
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Multiplication Property of Equality
Let a, b, and c represent numbers and let c ≠ 0. If a = b, then
a • c = b • c and
In other words, both sides of an equation may be multiplied or divided by the same nonzero number without changing the solution of the equation.
a b=
c c
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1313
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Solve for x
4x = 8To solve the equation for x, notice that 4 is multiplied by x.
To get x alone, we divide both sides of the equation by 4 and then simplify.
4 8
4 4
x= 11∙∙xx = 2 or = 2 or xx = 2 = 2
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1414
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Check
To check, replace x with 2 in the original equation.
4x = 8 Original equation
4 • 2 = 8 Let x = 2.
8 = 8 True.
The solution is 2.
?