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Simple Techniques For Solving Algebraic Expressions: Module 2

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Simple Techniques For Solving Algebraic Expressions: Module 2

2.1 Objectives of this Presentation

At the end of this presentation, the learners should be able to:

  • Grouping positive and negative term  to simplify an Algebraic Expressions
  • Grouping like and unlike terms  to simplify an Algebraic Expressions
  • Solve simple Algebraic Expressions
  • Solve Word Problems by forming and simplifying Algebraic Expressions
  • Identify Practical application of Algebra in real life scenario

2.1 Grouping like and unlike terms to simplify an Algebraic Expressions

Example 1: Consider this expression 3x + 6x + 5

Solution1:

3x + 6x + 5 = 3x + 6x + 5 (collect like terms together)

                     = 9x + 5

Example 2: Consider this expression 10y – 3y – 5y + 9

Solution2:

10y – 3y – 5y + 9 = 10y – 3y – 5y + 9 (collect like terms together)

                          = 10y – 3y – 5y + 9

                          = 7y – 5y + 9

                           = 2y + 9

Example 3: Consider this expression a- 7b + 3c + 8b + 2a 

Solution 3:

a- 7b + 3c + 8b + 2a   = a- 7b + 3c + 8b + 2a 

= a + 2a – 7b + 8b + 3c (collect like terms together)

= 3a + b + 3c 

Example 4: Evaluate the expression 2(4x – 3)

Solution 4:

2(4x – 3) = 2(4x – 3) expand the bracket

                = 8x – 12

Example 5: Evaluate the expression 2(4x – 3) when x = 5.

Solution 5:

                    2(4x – 3) = 2(4x – 3) expand the bracket

                                   = 8x – 12

                 Substitute the value into the expression, when x = 5.

                                   = 8(5) – 12

                                   = 40 – 12

                                   = 28

Example 6: Evaluate the expression 3x + 2 – 7x -4 + 5x + 6 when x =1.

Solution 6:

3x + 2 – 7x -4 + 5x + 6 = 3x + 2 – 7x – 4 + 5x + 6   (collect like terms together)

= 3x – 7x + 5x + 2 – 4 + 6

= – 4x + 5x + 2 – 4 + 6

= x + 2 – 4 + 6

= x – 2 + 6

= x + 4

Substitute the value into the expression, when x = 1.

= 1 + 4

= 5

2.2 Solve simple Algebraic Expressions

Example 7: Solve for x in the equation 2x + 7 = 15.

Solution: To isolate x, subtract 7 from both sides of the equation:

          LHS   = RHS

       2x + 7 – 7 = 15 – 7

                  2x = 8

Now, divide both sides by 2 to solve for x:

               (2x)/2 = 8/2

                      x = 4

Therefore, the solution is x = 4.

       * Now we can cross check to know whether our answer is correct (this step is optional)

            LHS   = RHS

                           2x + 7 = 15

                           2(4) + 7 = 15

                           8 + 7 = 15

                             15 = 15

Therefore, our solution to the equation is correct, hence (LHS = RHS)

Example 8: Solve for x in the equation 3x – 6 = 15.

Solution: To isolate x, add 6 to both sides of the equation:

          LHS   = RHS

       3x + 6 – 6 = 15 + 6

                  3x = 21

Now, divide both sides by 3 to solve for x:

               (3x)/3 = 21/3

                      x = 7

Therefore, the solution is x = 7.

       * Now we can cross check to know whether our answer is correct (this step is optional)

            LHS   = RHS

                          3x – 6 = 15

                           3(7) – 6 = 15 + 6

                           21 = 15 + 6

                             21 = 21

Therefore, our solution to the equation is correct, hence (LHS = RHS)

2.3 Solve Word Problems by forming and simplifying Algebraic Expressions

Simple Linear Equation:

Example 9: Melissa has twice as many apples as Mirabel. If Mirabel ‘x’ apples, how many apples does Melissa have?

Solution 9:

 Let ‘x’ represent the number of apples that Mirabel has. = (x)

Melissa has twice as many, so she has 2x apples. = (2x)

 Therefore both of them have = x + 2x (apples)

Solving for an Unknown:

Example 10: The sum of two consecutive odd integers is 28. What are the integers?

Solution 10:

Let ‘x’ represent the first odd integer. = ‘x’

The next consecutive odd integer would be’x + 2′. = ‘x + 2’.

The sum is 28,

So we can write the equation: x + (x + 2) = 28.

     x + x + 2 = 28.

        2x + 2 = 28.

To isolate x, subtract 2 from both sides of the equation

     LHS   = RHS

   2x + 2 – 2 =ss 28 – 2.

              2x = 26.

Now, solve for ‘x’ by dividing both side by 2

      2x/2   =  26/2.

            x = 13

       * Now we can cross check to know whether our answer is correct (this step is optional)

                                LHS = RHS

                                           x + (x + 2) = 28.

                        13 + (13 +2) = 28

                          13 + 15 = 28

                                 28 =28

Example 11: A woman is five time as old as her daughter, if the sum of their ages is 48 years. How old is the woman?

Solution 11:

Let ‘x’ represent the age of the daughter. = ‘x’

And the woman is 5 time the age of her daughter would be’5*x’. = ‘5x’.

The sum of their ages is 48,

So we can write the equation: x + (5x) = 48.

     x + 5x = 48. (Collect like terms together)

        6x = 48.

     LHS   = RHS

                 Now, solve for ‘x’ by dividing both side by 2

              6x = 48.

              6x/6 = 48/6

                    x = 8

Therefore, the daughter’ age x = 8 years,

and the woman whose is five times the daughter age 5x =  5*8 = 40

then sum of their ages = 8 + 40 = 48

2.4 Benefits of solving Algebraic Expressions

Solving algebraic expressions is not only a fundamental mathematical skill but also a valuable tool for enhancing problem-solving abilities, logical thinking, and career opportunities in various fields. It equips you with the quantitative literacy needed to navigate the complexities of our data-driven world, such as:

  1. Problem Solving: Algebraic expressions are used to solve real-world problems, making them a valuable tool in various fields like science, engineering, economics, and more. By solving these expressions, you can find solutions to practical problems.
  2. Mathematical Understanding: Solving algebraic expressions deepens your understanding of mathematical concepts and relationships. It helps you grasp how numbers and variables interact in equations and expressions.
  3. Critical Thinking: Algebra requires logical thinking and problem-solving skills. When you solve algebraic expressions, you develop critical thinking abilities, including pattern recognition, logic, and deduction.
  4. Preparation for Advanced Math: Algebra serves as a foundation for more advanced mathematical topics, including calculus, linear algebra, and statistics. Proficiency in algebraic manipulation is crucial for success in these higher-level courses.
  5. Career Opportunities: Many careers require at least a basic understanding of algebra. Proficiency in algebra can open doors to various professions in STEM (science, technology, engineering, and mathematics) fields and beyond.
  6. Problem-Solving Strategies: Algebraic problem-solving involves breaking down complex problems into simpler, solvable parts. These problem-solving skills are transferable and can be applied to a wide range of situations.
  7. Increased Confidence: As you become more skilled at solving algebraic expressions, your confidence in your mathematical abilities grows. This self-assurance can extend to other areas of life, where problem-solving skills are valuable.
  8. Academic Success: Algebra is a fundamental part of mathematics education. Success in algebra can boost your overall academic performance and provide a strong foundation for future mathematical studies.

2.5 Application of Algebra in Real Life

Algebraic expression has numerous applications in real life across various fields such as:

Personal Finance

  1. Budgeting: Algebra is used to create and adjust budgets, helping individuals allocate their income to expenses, savings, and investments.
  2. Interest Calculations: Algebraic formulas are used to calculate compound interest, helping people understand how their savings and investments grow over time.

Engineering

  1. Structural Analysis: Engineers use algebraic equations to analyze and design structures, such as bridges and buildings, to ensure they can withstand loads and stresses.
  2. Electrical Circuits: Algebra is used to model and analyze electrical circuits, allowing engineers to design and optimize electronic devices.

Physics

  1. Projectile Motion: Algebra is used to describe the path of a projectile, such as a thrown baseball or a launched rocket, and predict its trajectory.
  2. Force Equations: Algebraic equations like Newton’s second law (F = ma) are used to describe and analyze the behavior of objects under the influence of forces.

Business and Economics

  1. Market Analysis: Algebraic models are used to analyze market trends, forecast demand, and optimize pricing strategies.
  2. Financial Analysis: Algebraic formulas help businesses calculate profitability, break-even points, and return on investment (ROI).

Medicine

  1. Pharmacokinetics: Algebraic models are used to describe how drugs are absorbed, distributed, metabolized, and eliminated from the body, helping determine appropriate dosages.
  2. Medical Imaging: Algebraic algorithms are used in medical imaging techniques like MRI and CT scans to reconstruct images from raw data.

Computer Science

  1. Programming: Algebraic concepts are used in programming languages for tasks such as data manipulation, algorithm development, and optimization.
  2. Cryptography: Algebraic structures like modular arithmetic are fundamental to encryption algorithms used in data security.

Statistics

  1. Data Analysis: Algebra is used to perform statistical calculations, such as finding means, medians, standard deviations, and regression analysis, to interpret and draw conclusions from data.
  2. Statistics: Algebra is used in sports analytics to analyze player performance, develop game strategies, and make data-driven decisions.

Evaluate Yourself

Question 1: Solve these expressions

  1. -4x + 6x + 5 -8
  2. 3v + 6w + 5v – 8w +12
  3. 13e  + 7f + 9 – 5e – 8f +11

Question 2: Evaluate these expressions when x = 5.

  1. 2(4x – 7)
  2. 3(4x – 12)
  3. 2(3x + 7)

Question 3: World problems

  1. The sum of three consecutive odd integers is 21. What are the integers?
  2. The sum of four consecutive even integers is 28. What are the integers?
  3. A father is six time as old as his son, if the sum of their ages is 70 years. Find their respective ages?

Question 4:

  1. Highlight five benefits (5) of solving Algebraic Expressions
  2. Mentions five (5) Application of Algebra in Real Life

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Watch out for our next presentation on:

SIMPLE TECHNIQUES FOR SOLVING ALGEBRAIC EXPRESSIONS: MODULE 3


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