Simple Techniques For Solving Algebraic Expressions: Module 1

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1.1 Objectives of this Presentation

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

  • Define simple terms associated with  Algebraic Expressions
  • Differentiates between the  Expressions and Equations
  • State the coefficient of an Algebraic terms
  • Grouping positive and negative 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

1.2 Definition of Terms

1.2.1 Expression

 An expression is a mathematical phrase that consists of variables (such as x, y, z etc.), constants (numeric such 1- 9 etc.), and operations (addition, subtraction, multiplication, division, etc.). They do not contain an equal sign (=)

Example of expressions:

  1. 2x + 3
  2. 6e + 3e – 9
  3. 5y – 7

1.2.2 Equations

An equation is a mathematical statement that contains an equal sign (=) and demonstrates a relationship of equality between two expressions or quantities. Equations are used to solve for unknown values. They express that two expressions are equal and are used to find the value(s) of the variable(s) that make the equation true.

Examples of equations:

  1. 2x + 3 = 7
  2. 3y^2 – 8 = 1
  3. a + b = 10

1.2.3 Difference between Expression and Equation

Expressions and Equations are both fundamental concepts in algebra, but they have distinct differences: The key difference between Expressions and Equations are presented in Table 1.1:

Table 1.1: Major Differences between Expression and Equation

1.An expression is a mathematical phrase that consists of variables, constants, operations and do not contain an equal sign (=).An equation is a mathematical statement that contains an equal sign (=) and demonstrates a relationship of equality between two expressions or quantities.
2..Expressions do not contain an equal sign (=), example: 5y – 7Equation does contain an equal sign (=)., example: 5y – 7 = 13
3.Expression does not have any  solutions that make the values of the variables true,Equations can have one or more solutions (values of the variables that make the equation true), and they are often used to model real-world problems or solve mathematical problems.
4.Expressions represent mathematical quantitiesEquations represent statements of equality
5.Expressions are used to represent values or perform calculations.Expressions are used to represent values or perform calculations.

1.3 Algebraic Expressions

 An algebraic expression is a mathematical phrase that contains variables (such as x, y, z etc.) constants (number such 1 -9 etc.), and mathematical operations (such as addition, subtraction, multiplication, division, exponentiation, etc.). These expressions are used to represent and describe relationships between quantities or to perform mathematical computations.

1.3.1 Variables

Variables are qualities that can change in values; usually letters of the alphabets (a-z, A- Z) are used to represent variables.

1.3.2 Constant

Constants are a numeric value that does not have a variable attached to itself

1.3.3 Coefficients

Coefficients are numerical quantizes before a variable  

Example 1: Consider this expression 9x + 5


From the above expression: x is the variable, 9 is the coefficient and 5 is the constant term

Example 2: Consider this expression -2y – 3


From the above expression: y is the variable, -2 is the coefficient and -3 is the constant term

Example 3: Consider this expression ¾ h – 8

From the above expression: h is the variable, ¾ is the coefficient and -8 is the constant term

Note: coefficients are not always whole numbers, sometime they could be fractions

1.4 Types of Algebraic Expression

1.4.1 Simple Linear Expression:

  1. 2x + 3
  2. 5y – 7

1.4.2 Quadratic Expression:

  1. x^2 + 2x + 1
  2. 3y^2 – 6y + 2

1.4.3 Expression with Fractions:

  1. (1/2)x + (3/4)
  2. (2/5)y – (1/3)

1.4.4 Expression with Multiple Variables:

  1. 3x + 2y
  2. 2a – b + 4c

1.4.5 Expression with Exponents:

  1. 4x^2 – 9y^3
  2. (a^2 + b^2)/(a + b)

1.4.6 Uses of Algebraic Expressions

  • Algebraic expressions are often used in solving equations,
  • Simplifying mathematical problems, and representing real-world situations in terms of mathematical relationships.
  • They provide a way to work with unknown values (variables) and perform operations on them to derive solutions or analyze patterns.

1.5 Steps for Solving Algebra Expressions

Solving algebraic expressions typically involves simplifying or evaluating the expression to find its numerical value or a simplified form. The steps for solving algebraic expressions can vary depending on the complexity of the expression, the general steps to follow are:

1.5.1 Understand the Expression: Read the expression carefully and identify the variables, constants, and mathematical operations involved. Ensure you understand what you are asked to find or simplify. Follow the Order of Operations (PEMDAS/BODMAS):

  • Perform calculations within parentheses/brackets first.
  • Evaluate exponents (powers and roots).
  • Perform multiplication and division from left to right.
  • Perform addition and subtraction from left to right.
  • Substitute Values for Variables (if applicable):

If the expression contains variables, substitute the given values for those variables.

If no values are given, leave the variables as they are, unless further simplification is possible.

1.5.2 Perform the Calculations: Use the order of operations to perform the calculations step by step. Pay attention to signs (+/-) and the rules for operations.

Simplify each part of the expression systematically.

1.5.3 Combine Like Terms (if applicable): If there are like terms (terms with the same variable and exponent), combine them by adding or subtracting their coefficients.

1.5.4 Evaluate the Expression (if necessary): If you have simplified the expression to a point where no variables remain, calculate the numerical value of the expression.

1.5.5 Check Your Work: Double-check your calculations to ensure accuracy. Be especially careful with signs and operations.

1.5.6 Express Your Final Answer: Write the final answer in a clear and organized manner. If you were asked to find a numerical value, present that value. If you were asked to simplify the expression, write it in its simplest form.

1.5.7 Verify the Solution (if possible): If applicable, you can verify your solution by plugging the values back into the original expression and confirming that it equals the expected result.

Note: These steps are a general guideline for solving algebraic expressions. The complexity of the expression will determine how many of these steps are necessary and how involved each step is. Practice and familiarity with algebraic rules will help you become more proficient at solving algebraic expressions.

1.6 Grouping Positive and Negative Terms

Example 3: Simplify the expression 3x + 2x.

Solution 3:

3x + 2x. = 3x + 2x.

               = 5x

Example 4: Simplify the expression 8y – 6y.

Solution 4:

8y – 6y. = 8y – 5y.

               = 3y

Example 6: Simplify the expression 3f – 8f – 2f + 16f -4f.

Solution 6:

3f – 8f – 2f + 16f -4f = 3f – 8f – 2f + 16f -4f.

                                = 3f + 16f -8f -2f -4m (collect positive and negative signs together)

                                = 19f- 8f -2f -4f

                                = 19f- 14f

Evaluate Yourself

Question 1: Define the following terms with examples

  1. Equation
  2. Expression
  3. Variable
  4. Constant
  5. Coefficient

Question 2: Write out the variable, coefficient and constant for these expressions

  1. 4y – 2
  2. ¾ t +  ¾
  3. -2g +12

Question 3: In a tabular form differentiate between these pairs of words

  1. Equation and Expression
  2. Variable and coefficient

Question 4: Simplify the following expressions

  1. 8y + 2y – 6y + 3y + 5y
  2. 3k + 5k – 12k + 8k + 15k – 4k + 5k
  3. 7p + 5p – 6p + 6p -11p

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