Section 2.4

Solving Linear Equations in One Variable Using the Addition-Subtraction Principle

2.4 Lecture Guide: Solving Linear Equations in One Variable Using the Addition-Subtraction Principle

Objective: Solve linear equations in one variable using the addition-subtraction principle.

Linear Equation in One Variable

Algebraically

A linear equation in one variable x is an equation that can be written in the form Ax B , where A and B are realconstants and 0A .

VerballyA linear equation in one variable is first degree in this variable.

Algebraic Example2 24x

1. Which of the following choices are linear equations in one variable?

(a)

(b)

(c)

(d)

3 1 10x

3 10x y

23 1 10x

3 1 10x

Addition-Subtraction Principle of EqualityVerballyIf the same number is added to or subtracted from both sides of an equation, the result is an equivalent equation.

AlgebraicallyIf a, b, and c are real numbers, then a b

a c b c

a c b c

is equivalent to

and to

Numerical Example

3 7

3 3 7 ______

______

x

x

x

is equivalent to

and to

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution of each equation.

2. 4 11x

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation.

3. 2 5x

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation.

4. 8 1x

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation.

5. 3 3x

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation.

6. 5 4 3m m

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation.

7. 3 2 7y y

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation.

8. 7 6 1a a

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation.

9. 7 6d d

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation.

10. 4 5 3 8x x

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation.

11. 5 4 4 9x x

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation.

12. 8 2 7 12x x

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 13. 3 6 2 6x x

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 14. 6 2 4 11 8x x

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 15. 5 8 2 2 3x x

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 16. 7 2 2 13 1x x

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 17. 3 2 1 5 1m m

Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 18. 2 2 5 3 2x x

Based on the limited variety of equations we have examined, a good strategy to solve a linear equation in one variable is:

1. Remove any ____________ symbols.

2. Use the addition-subtraction principle of equality to move all ____________ terms to one side.

3. Use the addition-subtraction principle of equality to move all ____________ terms to the other side.

Objective: Identify a linear equation as a conditional equation, and identity, or a contradiction.

There are three classifications of linear equations to be aware of: conditional equations, identities, and contradictions.

Each of the equations in problems 2-18 is called a __________________ __________________ because it is only true for certain values of the variable and untrue for other values.

The following table compares all three types of linear equations.

Conditional Equation, Identity, and a Contradiction

Conditional Equation

Verbally A conditional equation is true for some values of the variable and false for other values.

Algebraic Example 2 3x x

Answer: 3xThe only value of x that checks is 3x .

Numerical Example

Conditional Equation, Identity, and a Contradiction

Identity

Verbally An identity is an equation that is true for all values of the variable.

Algebraic Example

Answer: All real numbers.

Numerical Example 2x x x

All real numbers will check. x x is always 2x.

Conditional Equation, Identity, and a Contradiction

Contradiction

Verbally A contradiction is an equation that is false for all values of the variable.

Algebraic Example Numerical Example 3x x

Answer: No solution.

No real numbers will check because no real number is 3 greater than its own value.

If the solution process for solving a linear equation in one variable produces a unique solution, then the original equation is a conditional equation. If the solution process results in the variable disappearing from both sides of the equation, then the equation you are trying to solve is either a contradiction or an identity

Identify each contradiction or identity and write the solution of the equation.

19. 3 2 4 2x x x

Identify each contradiction or identity and write the solution of the equation.

20. 5 3 2 3 3x x x

Identify each contradiction or identity and write the solution of the equation.

21. 4 2 3 7 12x x x

Identify each contradiction or identity and write the solution of the equation.

22. 3 4 2 5 1x x x

Simplify vs SolveSimplify the expression in the first column by combining like terms, and solve the equation in the second column.23. 7 3 6 5x x 24. 7 3 6 5x x

Simplify vs SolveSimplify the expression in the first column by combining like terms, and solve the equation in the second column.25. 26. 3 4 2 8x x 3 4 2 8x x

Objective: Use tables and graphs to solve a linear equation in one variable.

The solution of a linear equation in one variable is an x-value that causes both sides of the equation to have the same value. To solve a linear equation in one variable using tables or graphs, let 1Y equal the left side of theequation and let 2Y equal the right side of the equation. Using a table of values, look for the ______-value where the two ______-values are equal. Using a graph, look for the ______-coordinate of the point of __________________ of the two graphs. Note that the solution of a linear equation in one variable is an x-value and not an ordered pair.

27. Use the table shown to determine the solution of the equation 3 1 2 6x x .

The x-value in the table at which the two y values are equal is ______.

Solution: _____________

Verify your result by solving 3 1 2 6x x algebraically.

28. Use the graph shown to determine the solution of the equation .

The point where the two lines intersect has an x-coordinate of ______.

Solution: _____________

Verify your result by solving algebraically. 2 1 2x x

2 1 2x x

2, 6, 1 by 5, 10, 1

29.

Solve each equation using a table or a graph from your calculator by letting

0.5 3 0.5 5x x

1Y equal the left side of the equation and

2Y equal the right side of the equation. See Calculator Perspective 2.4.1 for help.

Solution: ____________

30.

Solve each equation using a table or a graph from your calculator by letting 1Y equal the left side of the equation and

2Y equal the right side of the equation. See Calculator Perspective 2.4.1 for help.

Solution: ____________

2 3 3 5 2x x x

Once the viewing window has been adjusted so you can see the point of intersection of two lines, the keystrokes required to find that point of intersection are ______ ______ ______ ______ ______ ______. To view a graph in the standard viewing window, press ZOOM ______.

31. Examine what happens if we try solving

3 2 2 1x x x by entering the left side as 1Yand the right side as 2Y on a graphing calculator.

10,10,1 by 10,10,1

(a) Do the two graphs appear to intersect?

31. Examine what happens if we try solving

3 2 2 1x x x by entering the left side as 1Yand the right side as 2Y on a graphing calculator.

(b) Compare the values of 1Y and 2Yfor each x-value in the table. What do you observe?

31. Examine what happens if we try solving

3 2 2 1x x x by entering the left side as 1Yand the right side as 2Y on a graphing calculator.

(c) Now solve 3 2 2 1x x x algebraically. Is this equation a conditional equation, a contradiction, or an identity?

Translate each verbal statement into algebraic form.

32. Three times a number z.

Translate each verbal statement into algebraic form.

33. Three less than a number w.

Translate each verbal statement into algebraic form.

34. Seven less than six times a number a.

Translate each verbal statement into algebraic form.

35. Two times the quantity of eight less than a number x.

36. Write an algebraic equation for the following statement, using the variable m to represent the number, and then solve for m.

Verbal Statement: Five less than three times a number is equal to two times the sum of the number and three.

Algebraic Equation:

Solve this equation:

37. The perimeter of the parallelogram shown equals

19 a . Find a.

aa

8

8