Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• Formulate and simplify algebraic expressions.
• Solve simple equations using inverse operations.
• Check your solutions to equations.
• Translate word problems into equations and solve them.
• Understand the difference between an expression and an equation.

Lesson summary

Algebraic expressions and simple equations form the foundation for more advanced mathematics. Understanding them is crucial for problem-solving in various aspects of life, from budgeting your pocket money to calculating ingredients for a recipe. In South Africa, these skills are essential for understanding financial literacy, participating in business, and even analyzing data related to social issues. This week, we will build on our previous knowledge of variables and constants and delve deeper into creating and solving algebraic expressions and simple equations.

Lesson notes

2. 1. Algebraic Expressions An algebraic expression is a mathematical phrase that combines numbers (constants), variables, and operation symbols (+, -, ×, ÷). A variable is a symbol (usually a letter like x, y, or n) that represents an unknown number. A constant is a fixed numerical value.

Examples: `3x + 5` (3 times a number x, plus 5) `y - 2` (A number y, minus 2) `2a + 4b` (2 times a number a, plus 4 times a number b) `7` (A constant expression - it doesn't have a variable)

Key Terms: Coefficient: The coefficient is the number that multiplies the variable. In the term `3x`, the coefficient is

3. Like Terms: Like terms have the same variable raised to the same power. We can combine like terms to simplify expressions. For example, `2x` and `5x` are like terms, but `2x` and `2x²` are not like terms.

Simplifying Algebraic Expressions: To simplify an expression, we combine like terms. This involves adding or subtracting the coefficients of like terms.

Example 1: Simplify `2x + 5 + 3x - 1` Identify like terms: `2x` and `3x` are like terms; `5` and `-1` are like terms.

Group like terms together: `2x + 3x + 5 - 1` Combine like terms: `(2 + 3)x + (5 - 1)` Simplify: `5x + 4` Example 2: Simplify `4y - 2 + y + 6 - 3y` Identify like terms: `4y`, `y`, and `-3y` are like terms; `-2` and `6` are like terms.

Group like terms together: `4y + y - 3y - 2 + 6` Combine like terms: `(4 + 1 - 3)y + (-2 + 6)` Simplify: `2y + 4` 2.

2. Simple Equations An equation is a mathematical statement that shows that two expressions are equal. It contains an equals sign (=).

Examples: `x + 3 = 7` `2y - 1 = 5` `4a = 12` Solving Simple Equations: Solving an equation means finding the value of the variable that makes the equation true. We use inverse operations to isolate the variable on one side of the equation. Inverse operations "undo" each other. Addition and subtraction are inverse operations. Multiplication and division are inverse operations.

Example 1: Solve the equation `x + 5 = 9` We want to isolate x. To do this, we need to get rid of the `+ 5`. The inverse operation of adding 5 is subtracting

5. Subtract 5 from both sides of the equation to keep it balanced: `x + 5 - 5 = 9 - 5` Simplify: `x = 4` Check the solution: Substitute x = 4 back into the original equation: `4 + 5 = 9`. This is true, so our solution is correct.

Example 2: Solve the equation `3y = 12` We want to isolate y. To do this, we need to get rid of the `3` which is multiplying y. The inverse operation of multiplying by 3 is dividing by

3. Divide both sides of the equation by 3: `3y / 3 = 12 / 3` Simplify: `y = 4` Check the solution: Substitute y = 4 back into the original equation: `3 * 4 = 12`. This is true, so our solution is correct.

Example 3: Solve the equation `z - 7 = 2` We want to isolate z. To do this, we need to get rid of the `- 7`. The inverse operation of subtracting 7 is adding

7. Add 7 to both sides of the equation: `z - 7 + 7 = 2 + 7` Simplify: `z = 9` Check the solution: Substitute z = 9 back into the original equation: `9 - 7 = 2`. This is true, so our solution is correct.

Example 4: Solve the equation `a/4 = 5` We want to isolate a. To do this, we need to get rid of the `/ 4`. The inverse operation of dividing by 4 is multiplying by

4. Multiply both sides of the equation by 4: `(a/4) 4 = 5 4` Simplify: `a = 20` Check the solution: Substitute a = 20 back into the original equation: `20 / 4 = 5`. This is true, so our solution is correct. 2.

3. Word Problems Often, we encounter problems described in words that we need to translate into algebraic expressions or equations to solve.

Example: "Thando has x marbles. Sipho has 5 more marbles than Thando. How many marbles does Sipho have?" Solution: Sipho has x + 5 marbles. This is an algebraic expression representing the number of marbles Sipho has.

Example: "Nomusa has y sweets. She gives 3 sweets to her friend. She now has 7 sweets left. How many sweets did Nomusa have originally?" Let y represent the number of sweets Nomusa had originally. Nomusa gave away 3 sweets, so we subtract 3 from y: `y - 3` She now has 7 sweets, so we can write the equation: `y - 3 = 7` Solve for y: Add 3 to both sides: `y - 3 + 3 = 7 + 3` Simplify: `y = 10` Nomusa originally had 10 sweets.

Check the solution: Substitute y = 10 back into the original equation: `10 - 3 = 7`. This is true, so our solution is correct. 2.

4. Expressions vs Equations It is important to note the difference between expressions and equations. Expressions are just combinations of numbers, variables and operators like `3x+5`. They do not have an equals sign. Equations have an equals sign, relating two expressions to each other like `3x + 5 = 11`. Guided Practice (With Solutions)

Question 1: Simplify the expression: `6a + 2 - 4a + 3` Solution: Identify like terms: `6a` and `-4a` are like terms; `2` and `3` are like terms.