MATH 107

Section 1.1 & 1.4Solving Linear & Quadratic

Equations

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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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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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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Solve the equation: 4 7 13x

1 1Solve the equation: 1 3 3 2

3 4x x

Solve the equation: 3 1 2 3 3 2x x x x

2 3 1

Solve the equation: 1 3 3 1x x x x

3 6Solve the equation: 1

2 2

x

x x

In the United States we measure temperature in both

degrees Fahrenheit ( F) and degrees Celsius ( C).

5They are related by the formula 32 .

9What are the Fahrenheit temperatures corresponding to

Cel

C F

sius temperatures of 10 ,0 ,and 40 ?

UNITS!!!!!!

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.

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.

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?

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?

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?

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.

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?

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?

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?

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?

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.

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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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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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EXAMPLE 1 Solving a Quadratic Equation by Factoring.

Solve by factoring:

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EXAMPLE 2 Solving a Quadratic Equation by Factoring.

Solve by factoring: 23 2 .t t

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EXAMPLE 3 Solving a Quadratic Equation by Factoring.

Solve by factoring: 2 16 8 .x x

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Suppose u is any algebraic expression and d ≥ 0.

THE SQUARE ROOT PROPERTY

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EXAMPLE 4 Solving an Equation by the Square Root Method

Solve: 23 5.x

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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 acx

a

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EXAMPLE 7 Solving a Quadratic Equation by Using the Quadratic Formula

Solve by using the quadratic formula. 23 5 2x x

Solution

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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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