The document provides procedures and examples for solving linear and quadratic equations. It discusses the steps to solve linear equations, which include eliminating fractions, simplifying, isolating the variable term, and checking the solution. For quadratic equations, it outlines the procedures of writing the equation in standard form, factoring the non-zero side, setting each factor equal to 0, and solving the resulting equations. Examples are provided to demonstrate solving quadratic equations by factoring, using the square root property, and applying the quadratic formula.

1. MATH 107
Section 1.1 & 1.4
Solving Linear & Quadratic
Equations

4. PROCEDURE FOR SOLVING LINEAR
EQUATIONS IN ONE VARIABLE
Step 1 Eliminate Fractions. Multiply both sides
of the equation by the least common
denominator (LCD) of all the fractions.
Step 2 Simplify. Simplify both sides of the
equation by removing parentheses and
other grouping symbols (if any) and
combining like terms.
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5. PROCEDURE FOR SOLVING LINEAR
EQUATIONS IN ONE VARIABLE
Step 3 Isolate the Variable Term. Add
appropriate expressions to both sides, so
that when both sides are simplified, the
terms containing the variable are on one
side and all constant terms are on the other
side.
Step 4 Combine Terms. Combine terms
containing the variable to obtain one term
that contains the variable as a factor.
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6. PROCEDURE FOR SOLVING LINEAR
EQUATIONS IN ONE VARIABLE
Step 5 Isolate the Variable. Divide both sides by
the coefficient of the variable to obtain the
solution.
Step 6 Check the Solution. Substitute the
solution into the original equation.
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7. Solve the equation: 4 7 13x − =

8. ( ) ( )
1 1
Solve the equation: 1 3 3 2
3 4
x x− − = +

9. ( ) ( ) ( ) ( )Solve the equation: 3 1 2 3 3 2x x x x− + = + −

10. ( ) ( )
2 3 1
Solve the equation:
1 3 3 1x x x x
= +
− + + −

11. 3 6
Solve the equation: 1
2 2
x
x x
+ =
− −

12. ( )
In the United States we measure temperature in both
degrees Fahrenheit ( F) and degrees Celsius ( C).
5
They are related by the formula 32 .
9
What are the Fahrenheit temperatures corresponding to
Cel
C F
° °
= −
sius temperatures of 10 ,0 ,and 40 ?− ° ° °

13. UNITS!!!!!!

14. A total of $16,000 is invested, some in stocks and some in bonds.
If the amount invested in bonds is one fourth that invested in
stocks, how much is invested in each category?
We are being asked to find the amount of two investments. These
amounts total $16,000.

15. A total of $16,000 is invested, some in stocks and some in bonds.
If the amount invested in bonds is one fourth that invested in
stocks, how much is invested in each category?
If x equals the amount invested in stocks, then the rest of the
money, 16,000 – x, is the amount invested in bonds.

16. A total of $16,000 is invested, some in stocks and some in bonds.
If the amount invested in bonds is one fourth that invested in
stocks, how much is invested in each category?

17. A total of $16,000 is invested, some in stocks and some in bonds.
If the amount invested in bonds is one fourth that invested in
stocks, how much is invested in each category?

18. A total of $16,000 is invested, some in stocks and some in bonds.
If the amount invested in bonds is one fourth that invested in
stocks, how much is invested in each category?

19. Andy grossed $440 one week by working 50 hours. His employer
pays time-and-a-half for all hours worked in excess of 40 hours.
What is Andy’s hourly wage?
We are looking for an hourly wage. Our answer will be expressed
in dollars per hour.

20. Andy grossed $440 one week by working 50 hours. His employer
pays time-and-a-half for all hours worked in excess of 40 hours.
What is Andy’s hourly wage?

21. Andy grossed $440 one week by working 50 hours. His employer
pays time-and-a-half for all hours worked in excess of 40 hours.
What is Andy’s hourly wage?

22. Andy grossed $440 one week by working 50 hours. His employer
pays time-and-a-half for all hours worked in excess of 40 hours.
What is Andy’s hourly wage?

23. Andy grossed $440 one week by working 50 hours. His employer
pays time-and-a-half for all hours worked in excess of 40 hours.
What is Andy’s hourly wage?

24. QUADRATIC EQUATION
A quadratic equation in the variable x is an
equation equivalent to the equation
where a, b, and c are real numbers and
a ≠ 0.
ax2
+ bx + c = 0,
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25. THE ZERO-PRODUCT
PROPERTY
Let A and B be two algebraic expressions.
Then AB = 0 if and only if A = 0 or B = 0.
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26. Step 1 Write the given equation in standard form
so that one side is 0.
Step 2 Factor the nonzero side of the equation.
Step 3 Set each factor to 0.
Step 4 Solve the resulting equations.
SOLVING A QUADRATIC
EQUATION BY FACTORING
Step 5 Check the solutions in original equation
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27. EXAMPLE 1 Solving a Quadratic Equation by Factoring.
Solve by factoring: 2x2
+5x = 3.
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28. EXAMPLE 2 Solving a Quadratic Equation by Factoring.
Solve by factoring:
2
3 2 .t t=
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29. EXAMPLE 3 Solving a Quadratic Equation by Factoring.
Solve by factoring:
2
16 8 .x x+ =
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30. Suppose u is any algebraic expression and d ≥ 0.
THE SQUARE ROOT
PROPERTY
If u2
= d, then u = ± d.
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31. EXAMPLE 4 Solving an Equation by the Square Root Method
Solve: ( )
2
3 5.x − =
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32. The solutions of the quadratic equation
in the standard form ax2
+ bx + c = 0
with a ≠ 0 are given by the formula
THE QUADRATIC FORMULA
2
4
.
2
b b ac
x
a
− ± −
=
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33. EXAMPLE 7
Solving a Quadratic Equation by Using the Quadratic
Formula
Solve by using the quadratic formula.
2
3 5 2x x= +
Solution
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34. EXAMPLE 10 Partitioning a Building
A rectangular building whose depth (from the
front of the building) is three times its
frontage is divided into two parts by a
partition that is 45 feet from and parallel to
the front wall. Assuming the rear portion of
the building contains 2100 square feet, find
the dimensions of the building.
Solution
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