**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:

**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.**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.**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.**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.**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.**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.**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.**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**

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

**Engineering**

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

**Physics**

**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.**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**

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

**Medicine**

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

**Computer Science**

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

**Statistics**

**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.**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**

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

**Question 2:** Evaluate these expressions **when x = 5.**

- 2(4x – 7)
- 3(4x – 12)
- 2(3x + 7)

**Question 3: World problems**

- The sum of three consecutive odd integers is 21. What are the integers?
- The sum of four consecutive even integers is 28. What are the integers?
- 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:**

- Highlight five benefits (5) of solving Algebraic Expressions
- 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**