2. Linear Equations and Applications
Chapter 2
Linear Equations and Applications

3. Linear Equations in One Variable
2.1
Linear Equations in One Variable

4. 2.1 Linear Equations in One Variable
Objectives
Decide whether a number is a solution of a linear
equation.
Solve linear equations using the addition and
multiplication properties of equality.
Solve linear equations using the distributive
property.
Solve linear equations with fractions or decimals.
Identify conditional equations, contradictions,
and identities.
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5. 2.1 Using Linear Equations of One Variable
Algebraic Expressions vs. Equations
In the previous chapter, we looked at algebraic expressions:
– 9y + 5, k, and
Equations are statements that two algebraic expressions are equal:
3x – 13 = 29, y = – 11, and 3m = 4m – 2
An equation always contains an equals sign, but an expression does not.
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6. 2.1 Using Linear Equations of One Variable
Linear Equations in One Variable
Linear Equation in One Variable
A linear equation in one variable can be written in the form
Ax + B = C
where A, B, and C are real numbers, with A = 0. /
A linear equation is also called a first-degree equation since the greatest power on the variable is one.
5x + 10 = 13
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7. 2.1 Using Linear Equations of One Variable
Linear Equations in One Variable
Determine whether the following equations are linear or nonlinear.
8x + 3 = –9
Yes, x is raised to the first power.
No, x is not raised to the first power.
9x3 – 8 = 15 7
No, x is not raised to the first power.
= –12 x
No, x is not raised to the first power.
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8. 2.1 Using Linear Equations of One Variable
Deciding Whether a Number is a Solution
If a variable can be replaced by a real number that makes the equation a true statement, then that number is a solution of the equation, x – 10 = 3.
13 is a solution
8 is not a solution 13 8
x – 10 = 3
x – 10 = 3
13 – 10 = 3
(true)
8 – 10 = 3
(false)
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9. 2.1 Using Linear Equations of One Variable
Finding the Solution Set of an Equation
An equation is solved by finding its solution set – the set of all solutions.
The solution set of
x – 10 = 3
is {13}.
Equivalent equations are equations that have the same solution set. These are equivalent equations since they all have solution set {–3}.
3x + 5 = –4
3x = –9
x = –3
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10. 2.1 Using Linear Equations of One Variable
Solving Linear Equations
An equation is like a balance scale, comparing the weights of two quantities.
Expression-1 =
Expression-2
We apply properties to produce a series of simpler equivalent equations to determine the solution set.
Variable =
Solution
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11. 2.1 Using Linear Equations of One Variable
Addition Property of Equality
The same number may be added to both sides of an equation without changing the solution set.
A = B A = B A + C = B + C
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12. 2.1 Using Linear Equations of One Variable
Multiplication Property of Equality
Each side of an equation may be multiplied by the same nonzero number without changing the solution set.
A = B A = B A C = B C
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13. 2.1 Using Linear Equations of One Variable
Addition and Multiplication Properties of Equality
Addition Property of Equality
For all real numbers A, B, and C, the equation
A = B and A + C = B + C
are equivalent.
Multiplication Property of Equality
For all real numbers A, B, and for C = 0, the equation
A = B and A C = B C
are equivalent. /
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14. 2.1 Using Linear Equations of One Variable
Addition and Multiplication Properties of Equality
Because subtraction and division are defined in terms of addition and multiplication,
we can extend the addition and multiplication properties of equality as follows:
The same number may be subtracted from each side of an equation, and each side of an equation may be divided by the same nonzero number, without changing the solution set.
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15. 2.1 Using Linear Equations of One Variable
Solving Linear Equations in One Variable
Step 1
Clear fractions. Eliminate any fractions by multiplying each side by the least common denominator.
Step 2
Simplify each side separately. Use the distributive property to clear parentheses and combine like terms as needed.
Step 3
Isolate the variable terms on one side. Use the addition property to get all terms with variables on one side of the equation and all numbers on the other.
Step 4
Isolate the variable. Use the multiplication property to get an equation with just the variable (with coefficient of 1) on one side.
Step 5
Check. Substitute the proposed solution into the original equation.
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16. 2.1 Using Linear Equations of One Variable
Solving Linear Equations
Solve 3x + 2 = 10.
3x + 2 = 10
3x + 2 – 2 = 10 – 2
Subtract 2.
3x = 8
Combine like terms.
Divide by 3.
Proposed solution.
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17. 2.1 Using Linear Equations of One Variable
Solving Linear Equations
3x + 2 = 10 8
Check by substituting the proposed solution back into the original equation.
3 • = 10 3
Since the value of each side is 10, the proposed solution is correct.
8 + 2 = 10
The solution set is .
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18. 2.1 Using Linear Equations of One Variable
Solving Linear Equations
Solve 2x – 5 = 5x – 2.
2x – 5 = 5x – 2
2x – 5 – 5x = 5x – 2 – 5x
Subtract 5x.
–3x – 5 = –2
Combine like terms.
–3x – 5 + 5= –2 + 5
Add 5.
–3x = 3
Combine like terms.
Divide by –3.
x = –1
Proposed solution.
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19. 2.1 Using Linear Equations of One Variable
Solving Linear Equations
Check by substituting the proposed solution back into the original equation.
2x – 5 = 5x – 2
2(–1) – 5 = 5(–1) – 2
Since the value of each side is –7 , the proposed solution is correct.
–2 – 5 = –5 – 2
–7 = –7
The solution set is {–1}.
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20. 2.1 Using Linear Equations of One Variable
Solving Linear Equations
Solve 5(2x + 3) = 3 – 2(3x – 5).
5(2x + 3) = 3 – 2(3x – 5)
10x + 15 = 3 – 6x + 10
10x + 15 – 15 = 3 – 6x + 10 – 15
10x = – 6x – 2
10x + 6x = –6x – 2 + 6x
16x = –2
Distributive Prop.
Add –15.
Collect like terms.
Add 6x.
Collect like terms.
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21. 2.1 Using Linear Equations of One Variable
Solving Linear Equations
16x = –2
Divide by 16.
Proposed solution.
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22. 2.1 Using Linear Equations of One Variable
Solving Linear Equations
Check proposed solution:
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23. 2.1 Using Linear Equations of One Variable
Solving Linear Equations with Fractions
Solve .
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24. 2.1 Using Linear Equations of One Variable
Solving Linear Equations with Fractions
continued
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25. 2.1 Using Linear Equations of One Variable
Solving Linear Equations with Decimals
Solve .
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26. 2.1 Using Linear Equations of One Variable
Solving Linear Equations with Decimals
continued
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27. 2.1 Using Linear Equations of One Variable
Conditional, Contradiction, and Identity Equations
Linear equations can have exactly one solution, no solution, or an infinite number of solutions.
Type of Linear Equation
Number of Solutions
Indication When Solving
Conditional
One
Final results is x = a number.
Identity
Infinite; solution set {all real numbers}
Final line is true, such as 5 = 5.
Contradiction
None; solution set is
Final line is false, such as –3 = 11.
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28. 2.1 Using Linear Equations of One Variable
Conditional, Contradiction, and Identity Equations
A contradiction has no solutions.
Since 0 = –5 is never true, and this equation is equivalent to x + 7 = x + 2, the solution set is empty.
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29. 2.1 Using Linear Equations of One Variable
Conditional, Contradiction, and Identity Equations
An identity has an infinite number of solutions.
Since 0 = 0 is always true, and this equation is equivalent to 2x + 2 = 2(x + 1), the solution set is all real numbers.
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