Hawkes Learning Systems: College Algebra

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Hawkes Learning Systems:College Algebra

Section 2.1a: Linear Equations in One Variable




Objectives

o Equivalent equations and the meaning of solutions.o Solving linear equations. o Solving absolute value equations.




Equations and the Meaning of Solutions

o An equation is a statement that two algebraic expressions are equal.

o To solve an equation means to find the solution(s): value(s) of the variable that make the equation true.

o The set of all such values is called the solution set.




Types of Equations

There are three types of equations:

1. A conditional equation has a countable number of solutions. For example, x + 7 = 12

has one solution, 5. The solution set is {5}.

2. An identity is true for all real numbers and has an infinite number of solutions. For example,

is true for all real number values of . The solution set is R.

3. A contradiction is never true and has no solution. For example, is false for any value of . The solution set is Ø.

x6x x

x

2( 1)x x x x




Linear Equations in One Variable

A linear equation in one variable, such as the variable , is an equation that can be transformed into the form,

where and are real numbers and . Such equations are also called first-degree equations, as appears to the first power.

x

0ax b

a b 0a

x




Linear Equations and Equivalent Equations

o We solve linear equations by performing the same operations on both sides of the equation.

o This results in simpler equivalent equations that are easier to solve and have the same solution.




Solving Linear Equations

To solve a linear equation (in x): 1. Simplify each side of the equation separately by

removing any grouping symbols and combining like terms. 2. Add or subtract the same expression(s) on both sides of the

equation in order to get the variable term(s) on one side and the constant term(s) on the other side of the equation and simplify.

3. Multiply or divide by the same nonzero quantity on both sides of the equation in order to get the numerical coefficient of the variable term to be one.

4. Check your answer by substitution in the original equation.




Example: Solving Linear Equations

Step 1: Simplify

Step 2: Add or Subtract

Step 3: Multiply or Divide

3 5 4 7x x x

3 5 5 7x x

12 2x

6 x

Solve:





The solution set is R.

2( 7) 14x x x

2 14 14x x x

Solve:





The solution set is R.

6 7 22 12 8

4 8 8

y y Solve:





No Solution

0.03 0.67( 5) 0.7 3.44w w w

0.03 0.67 3.35 0.7 3.44w w w

0.7 3.35 0.7 3.44w w

Solve:

The solution set is Ø.




Example: Solving Linear EquationsSolve.




Solving Absolute Value Equations

The absolute value of any quantity is either the original quantity or its negative (opposite). This means that, in general, every occurrence of an absolute value term in an equation leads to two equations with the absolute value signs removed, if c > 0. Note: if c < 0, it has no solution.

ax b c means

ax b c or ax + b = -c




Example: Solving Absolute Value Equations

Step 1: Rewrite the absolute value equation without absolute values.

3 2 5x

3 2 5x

3 7x

7

3x 1x

Step 2: Solve the two equations

or 3x – 2 = -5

or 3x = -3

or

Solve:




Example: Absolute Value Equations

4 3 2 0x Solve:

|4x + 3| = -2 False, absolute value is never negative. No solution; the solution set is Ø.

Solve: |6x – 2| = 06x – 2 = 06x = 2x = ⅓

If |ax + b| = 0, then ax + b = 0.




Example: Absolute Value EquationsSolve.

|x – 4| = |2x + 1|

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Hawkes Learning Systems: College Algebra