Solving for a Variable

Site: St. Louis
Course: Michigan Algebra I Sept. 2012
Book: Solving for a Variable
Printed by: Guest user
Date: Thursday, November 21, 2024, 11:43 AM

Description

Addition & Subtraction

The process of solving one-step equations involving addition or subtraction, such as x + 6 = 3, is vital to understanding how to work with linear equations. To review how to solve for a variable, select the following link:

Review Solving One-Step Equations with Addition or Subtraction


Guided Practice

To solidify your understanding of solving one-step equations, visit the following link to Holt, Rinehart and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.

Guided Practice

Multiplication & Division

The process of solving one-step equations involving multiplication or division, such as 2x = 5, is a necessary skill in solving linear equations. To review how to solve for a variable, select the following link:

Review Solving Equations Using Multiplication or Division


Example 1

Solve Solve_ex1_1

Step 1. Since the x is divided by 5, multiply both sides by 5.

Solve_ex1_2

Step 2. Check the solution by substituting into the original equation.

Solve_ex1_3

Example 2

Solve Solve_ex2_1 .

Step 1. Since x is multiplied by Solve_ex2_1.2 , divide both sides by Solve_ex2_1.2 which is the same as multiplying both sides by its reciprocal of Solve_ex2_1.2 .

Solve_ex2_2

Step 2. Simplify.

Solve_ex_2_3

Step 3. Check the solution by substituting into the original equation.

Solve_ex2_4


Guided Practice

To solidify your understanding of solving equations using multiplication and division, visit the following link to Holt, Rinehart and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.

Guided Practice

Multi-Step Equations

In multi-step equations, the same techniques used in solving one-step equations will be used. Just like solving one-step equations, the goal is to have the variable with a coefficient of one on one side of the equal sign and a number on the other. Keep in mind that the variable does not always have to be x. Equations can make use of any letter as a variable.

The strategy for getting the variable by itself with a coefficient of 1 involves using opposite operations. Many linear equations require more than one step for their solution.


Example 3

Solve 7x + 2 = -54

Step 1. Subtract 2 from both sides of the equation to isolate the variable.

Solve_ex3_1

Step 2. Divide both sides of the resulting equation by the coefficient of 7 and simplify.

Solve_ex3_2

Step 3. Check the solution.

Solve_ex3-3

*Hint: When solving a multi-step equation follow the Order of Operations in

reverse order or SADMEP.

Example 4

Solve -5x - 7 = 108

Step 1. Add 7 to both sides of the equation to isolate the variable.

Solve_ex4_1

Step 2. Divide both sides of the resulting equation by -5 and simplify.

Solve_ex4_2

Step 3. Check the solution.

Solve_ex4_3

Example 5

Solve Solve_ex5


Step 1. Subtract 7 from both sides of the equation to isolate the variable.


Solve_ex5_1


Step 2. Divide both sides of the resulting equation by Solve_ex5_2-1 and simplify. *Remember: multiplying both sides by the reciprocal of the coefficient yields the same result as dividing by the coefficient. Choose the easiest method.

Solve_ex5_2

Step 3. Check the solution.

Solve_ex5_3


Combining Variables

In solving equations in the form ax + c = bx + d the goal remains to rewrite the equation in the form variable = constant. Begin by combining like terms so there is only one variable term and one constant term.

Example 6 Solve 5x + 7x = 72

Step 1. Combine like terms.

5x + 7x = 72
12x = 72

Step 2. Divide both sides of the resulting equation by 12 and simplify.

12x= 72
12 12

x = 6

Step 3. Check the solution.

5?6 + 7?6 = 72
30 + 42 = 72
72 = 72

Example 7

Solve 4x - 6 = 6x

Step 1. Use inverse functions to move like terms to the same side of the equation. Subtract 4x from both sides and simplify.

Solve_ex7_1

Step 2. Divide both sides of the resulting equation by 2 and simplify.

Solve_ex7_2

Step 3. Check the solution.

Example 8

Solve 8x - 1 = 23 - 4x

Step 1. Use inverse functions to move like terms to the same side of the equation. Add 4x to both sides and simplify.

Solve_ex8_1

Step 2. Add 1 to both sides of the resulting equation and simplify.

Solve_ex8_2

Step 3. Divide both sides of the resulting equation by 12 and simplify.

Solve_ex8_3

Step 4. Check the solution.

Example 9

Solve 5 + 4x - 7 = 4x - 2 - x

Step 1. Combine like terms that are on the same side of the equation.

Solve_ex9_1

Step 2. Use inverse functions to move like terms to the same side of the equation. Subtract 3x from the resulting equation.

Solve_ex9_2

Step 3. Add 2 to both sides of the resulting equation.

Solve_ex9_3

Step 4. Check the solution.


Solve_ex9_4

Special Cases

There are two special cases that may arise when solving equations. They occur when the variable is eliminated from the equation, leaving a constant on both sides. If the resulting statement is true, such as 2 = 2, then any real number can be substituted for the variable and a true statement will be created. Therefore, the solution is all real numbers.

The other case occurs when the statement is false, such as 6 = 7, then any real number substituted in for the variable will create a false conclusion. Therefore, there is no solution.


Example 10

Solve 11 + 3x - 7 = 6x + 5 - 3x

Step 1. Combine like terms that are on the same side of the equation.

Solve_ex10-1

Step 2. From the resulting equation, subtract 3x from both sides.

Solve_ex10_2

Since 4 = 5 is a false statement, there is no solution.

Example 11

Solve 6x + 5 - 2x = 4 + 4x + 1

Step 1. Combine like terms that are on the same side of the equation.

Solve_ex11_1

Step 2. From the resulting equation, subtract 4x from both sides.

Solve_ex11_2

Since 5 = 5 is a true statement, all real numbers are solutions.

Distribution Property

When an equation contains parentheses, one of the steps in solving the equation requires the use of the Distributive Property. The Distributive Propertyis used to remove the parentheses from a variable expression by multiplying every term within the parentheses by the value in front of the parentheses.

Example 12

Solve 9 = 3(5x - 2)

Step 1. Use the Distributive Property to remove the parentheses.

Solve_ex12_1

Step 2. Combine like terms in the resulting equation. Add 6 to both sides of the equation.

Solve_ex12_2

Step 3. Solve for x. Divide both sides of the resulting equation by 15 and simplify.

Solve_ex12_3

Step 4. Check the solution.


Solve_ex12_4

Example 13

Solve 6x - (3x + 8) = 16

Solve_ex13_1

Solution Check

6?8 - (3?8 +8) = 16
48 - 32 = 16
16 = 16

Example 14

Solve 7(5x - 2) = 6(6x - 1)

Solve_ex14_1

Solution Check

7(5? (-8) - 2) = 6(6?(-8) - 1)
7(-42) = 6(-49)
-294 = -294

Example 15

Solve 13 - (2x + 2) = 2(x + 2) + 3x

Solve_ex15

Solution Check

13 - (2?1+2) = 2(1+2) + 3?1
13 - 4 = 6 + 3
9 = 9

Video Lesson

To learn how to solve for a variable, select the following link:

Solving Multi-Step Equations

Guided Practice

To solidify your understanding of solving for variables, visit the following link to Holt, Rinehart and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.

Guided Practice

Practice

Solving Linear Equations Worksheet

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Answer Key

Solving Linear Equations Answer Key

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Sources

Holt, Rinehart, & Winston. "Algebraic Reasoning. "http://my.hrw.com/math06_07/nsmedia/homework_help/msm3/msm3_ch01_11_homeworkhelp.html (accessed August 15, 2010)

Holt, Rinehart, & Winston. "Algebraic Reasoning. "http://my.hrw.com/math06_07/nsmedia/homework_help/msm3/msm3_ch01_08_homeworkhelp.html (accessed August 15, 2010)

Holt, Rinehart, & Winston. "Algebraic Reasoning."
http://my.hrw.com/math06_07/nsmedia/homework_help/msm3/msm3_ch11_03_homeworkhelp.html (accessed August 15, 2010)

Mainland High School, Algebra Lab. "Solving Multi-Step Equations." http://www.algebralab.org/lessons/lesson.aspx?file=algebra_onevariablemultistep.xml (accessed 08/21/2010).

Stapel, Elizabeth. "Solving Multi-Step Linear Equations." Purplemath. Available from http://www.purplemath.com/modules/solvelin3.htm . Accessed 15 August 2010