Transcript Section 2.1

Chapter 2
Linear Equations and Inequalities
in One Variable
§ 2.1
The Addition Property of Equality
The Addition Property of Equality
The same real number (or algebraic
expression) may be added to both sides of an
equation without changing the equation’s
solution.
This can be expressed symbolically as follows:
If a = b, then a + c = b + c
Blitzer, Introductory Algebra, 5e – Slide #3 Section 2.1
Linear Equations
Definition of a Linear Equation
A linear equation in one variable x is an
equation that can be written in the form
ax + b = 0, where a and b are real numbers
and a is not equal to 0.
An example of a linear equation in x is 4x + 2 = 6. Linear
equations in x are first degree equations in the variable x.
Blitzer, Introductory Algebra, 5e – Slide #4 Section 2.1
Properties of Equality
Property
Definition
Addition Property of Equality
The same real number or algebraic
expression may be added to both sides
of an equation without changing the
equation’s solution set.
Subtraction Property of Equality
The same real number or algebraic
expression may be subtracted from
both sides of an equation without
changing the equation’s solution set.
Blitzer, Introductory Algebra, 5e – Slide #5 Section 2.1
Solving Linear Equations
Solving a Linear Equation
1) Simplify the algebraic expressions on each side.
2) Collect all the variable terms on one side and all the
numbers, or constant terms, on the other side
3) Isolate the variable and solve.
4) Check the proposed solution in the original equation.
Blitzer, Introductory Algebra, 5e – Slide #6 Section 2.1
Solving an equation using the addition property
Solve the equation : x  9  12
We can isolate the variable, x, by adding 9 to both sides.
x  9  12
x  9  9  12  9
x  21
The set of an equation' s solutions is called its
solution set. The solution is 21 or in set notation {21}.
Blitzer, Introductory Algebra, 5e – Slide #7 Section 2.1
Subtracting from both sides of an equation
Since we know that subtraction is just
addition of an opposite or an additive inverse,
we can also subtract the same number from
both sides of an equation without changing
the equation’s solution.
Solve: x + 6 = 9
x + 6 – 6 = 9 – 6 Subtract 6 from both sides.
x=3
Blitzer, Introductory Algebra, 5e – Slide #8 Section 2.1
Adding and Subtracting Variable Terms in an Equation
Now consider an equation in which we would need to
subtract variable terms from both sides. Remember that
our goal is to isolate all the variable terms on one side. To do
this in the equation below, we must get the 4x term off the
RHS by adding its opposite, -4x, to both sides.
Solve: 5x = 4x + 3
5x – 4x = 4x + 3 – 4x Subtract 4x from both sides.
This simplifies to: x = 3
Blitzer, Introductory Algebra, 5e – Slide #9 Section 2.1
Solving Linear Equations
EXAMPLE: Solve for x
14x + 2 = 15x
2) Collect variable terms on one side and constant terms
on the other side.
14x – 14x + 2 = 15x – 14x
2=x
Subtract 14x from both sides
Simplify
Check the proposed solution of 2 in the original equation. When we insert the 2 for x, we
Get the sentence 14(2) + 2 = 15(2) or we get 28 + 2 = 30, which is true. Then x = 2 is the
solution.
Blitzer, Introductory Algebra, 5e – Slide #10 Section 2.1
Solving Linear Equations
EXAMPLE
Solve and check: 5 + 3x - 4x = 1 - 2x + 12.
SOLUTION
1) Simplify the algebraic expressions on each side.
5 + 3x - 4x = 1 - 2x + 12
5 - x = 13 - 2x
Combine like terms:
+3x - 4x = -x
1 + 12 = 13
Blitzer, Introductory Algebra, 5e – Slide #11 Section 2.1
Solving Linear Equations
CONTINUED
2) Collect variable terms on one side and constant terms
on the other side.
5 - x + 2x = 13 - 2x + 2x
5 + 1x = 13
5 – 5 + 1x = 13 - 5
x=8
Add 2x to both sides
Simplify
Subtract 5 from both sides
Simplify
Blitzer, Introductory Algebra, 5e – Slide #12 Section 2.1
Solving Linear Equations
CONTINUED
4) Check the proposed solution in the original equation.
5 + 3x - 4x = 1 - 2x + 12
5 + 3(8) - 4(8) ?= 1 – 2(8) + 12
?
5 +24 – 32 = 1 – 16 + 12
29 -32 ?= – 16 + 13
-3 = -3
Blitzer, Introductory Algebra, 5e – Slide #13 Section 2.1
Original equation
Replace x with 8
Multiply
Add or subtract from left to
right
True - It checks. The
solution set is {8}.