LINEAR EQUATIONS IN ONE VARIABLE AND FORMULAS

SECTIONS 2.1 – 2.6

Directions:

  1. Read the explanations and examples.
  2. Copy the exercise problems and work them out on your own paper.
  3. Be sure to include your name, section number, and date.
  4. Fax or email your work to your instructor by the stated calendar date.

EQUATION-An equation is a statement indicating that two quantities are equal.

LINEAR EQUATION IN ONE VARIABLE- A linear equation in one variable can be written in the form

ax + b = 0 where x is the variable with an exponent of 1;

a and b are real numbers (a ¹ 0).

Example: 3x + 11 = 0

SOLVING LINEAR EQUATIONS IN ONE VARIABLE

To solve an equation means to find the value(s) of the variable for which the statement is true. Use the Addition and/or Multiplication Principle to solve the equation by rewriting the equation in the equivalent form x = a number.

Addition Principle

The same number can be added to both sides of an equation without changing the balance of the equation. (This rule is also true for subtracting the same number from both sides of an equation since subtraction is addition of the opposite.)

Multiplication Principle

Both sides of an equation can be multiplied by the same nonzero number without changing the balance of the equation. (This rule is also true for dividing both sides of an equation by the same nonzero number since division is multiplication by the reciprocal.)

SOLUTIONS TO LINEAR EQUATIONS IN ONE VARIABLE

A linear equation in one variable may have

  1. one solution – one value for the variable that makes the statement true;
  2. no solution – no real number that makes the statement true
  3. an infinite number of solutions – any real number makes the statement true.

See examples below.

Example 1: Solve 3x + 8 = 20

                                         3x = 12

                    One solution: x = 4

Check the solution: 3x + 8 = 20

                              3(4) + 8 = 20

                                12+ 8 = 20

                                       20 = 20a

Example 2: Solve 5x – 4 = 3x + 6.

                            2x – 4 = 6

                                2x = 10

One solution: x = 5

Check the solution: 5x – 4 = 3x + 6

                                5(5) – 4 = 3(5) + 6

    25 - 4 = 15 + 6

    21 = 21a

SOLVING A FORMULA FOR AN INDICATED LETTER

A formula is a statement that involves a relationship among two or more quantities. The equation for the formula contains two or more letters. Some of these letters represent variables and some represent constants. (For example: In the formula C = p d, the letters C and d represent variables, but the letter p represents a constant.)

A formula is said to be solved for a letter if that letter appears by itself on only one side of the formula, with a coefficient of 1.

To solve a formula for an indicated letter, use the Addition and Multiplication Principles to isolate the indicated letter (with a coefficient of 1) by itself on one side of the formula.

Exercises:

  1. 4x – 10 = 2

    2.  

  2. x = 15

    3.  

  3. (x – 4) = x

    4.  

  4. 5(2x + 3) = 3 – 2(3x – 5)

    5.  

  5. -(1 – 2x) = 3 + 2x

    6.  

    Solve each formula for the indicated letter.

  6. Solve for T: PV = NRT

    7.  

  7. Solve for W: P = 2L + 2W

    8.  

  8. Solve for b: a =

    9.  

  9. Solve for y: 3x + 6y = 12

 

10.   Solve for d: B = a + (c +3)d

Copyright 1999, Collin County Community College District and Rosemary M. Karr, Ph.D. All material on this site is for CCCCD class use only. Any unauthorized duplication or distribution is prohibited.