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

2. 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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3. 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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4. 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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5. 2.1 Using Linear Equations of One Variable
Solving Linear Equations in One Variable
Step 1
Clear fractions. Eliminate any fractions if possible 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 properties of equality to get all terms with variables on one side of the equation and all numbers on the other.
Step 4
Check. Substitute the proposed solution into the original equation.
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6. 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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7. 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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8. 2.1 Using Linear Equations of One Variable
Solving Linear Equations
Warm Up
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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9. 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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10. 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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11. 2.1 Using Linear Equations of One Variable
Solving Linear Equations
16x = –2
Divide by 16.
Proposed solution.
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12. 2.1 Using Linear Equations of One Variable
Solving Linear Equations
Check proposed solution:
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13. 2.1 Using Linear Equations of One Variable
Solving Linear Equations with Fractions
Solve .
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14. 2.1 Using Linear Equations of One Variable
Solving Linear Equations with Fractions
continued
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15. 2.1 Using Linear Equations of One Variable
Solving Linear Equations with Decimals
Solve .
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16. 2.1 Using Linear Equations of One Variable
Solving Linear Equations with Decimals
continued
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17. 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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18. 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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19. 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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20. Ticket-Out-The-Door How do you know
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