Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Simplify algebraic expressions by combining like terms.
• Identify the coefficient and variable in a term.
• Solve one-step linear equations using inverse operations.
• Formulate simple expressions from word problems.
• Verify the solution to an equation by substitution.

Lesson summary

Algebra is a fundamental building block in mathematics, opening doors to more complex problem-solving in subjects like science, engineering, and economics. This week, we delve into algebraic expressions and simple equations, focusing on simplifying expressions, identifying like terms, and solving equations using inverse operations. These skills are crucial for managing your finances, understanding scientific concepts, and even for everyday tasks like calculating discounts at the shops or sharing costs with friends. In South Africa, where resourcefulness and problem-solving are highly valued, a strong understanding of algebra is essential for success in various fields.

Lesson notes

2.1 Algebraic Expressions: An algebraic expression is a combination of variables, constants, and mathematical operations (addition, subtraction, multiplication, division, exponents, etc.).

Variable: A letter (e.g., x, y, z, a, b) that represents an unknown quantity. Think of it as a placeholder for a number we don't yet know.

Constant: A number that has a fixed value (e.g., 5, -3, 1/2).

Coefficient: The number multiplied by a variable in a term (e.g., in the term 3x, 3 is the coefficient). If there's no number explicitly written, the coefficient is assumed to be 1 (e.g., x has a coefficient of 1).

Like Terms: Terms that have the same variable raised to the same power (e.g., 3x and 5x are like terms; 2y 2 and -7y 2 are like terms; 4 and -9 are like terms). Constants are always like terms.

Simplifying Algebraic Expressions: Simplifying an algebraic expression means combining like terms. You can only add or subtract like terms.

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

Combine like terms: (2x - x) + (3y + 5y)

Simplify: x + 8y Example 2: Simplify 5a + 2b - 3a + 7 - b + 1 Identify like terms: 5a and -3a; 2b and -b; 7 and 1 Combine like terms: (5a - 3a) + (2b - b) + (7 + 1)

Simplify: 2a + b + 8 2.2 Simple Equations: An equation is a mathematical statement that shows that two expressions are equal. It contains an equals sign (=). A simple equation is one where we are solving for a single unknown variable.

Solving Simple Equations: To solve an equation, we want to isolate the variable on one side of the equals sign. We do this by using inverse operations. An inverse operation "undoes" another operation. The inverse of addition is subtraction. The inverse of subtraction is addition. The inverse of multiplication is division. The inverse of division is multiplication.

Important Rule: Whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain the balance. Think of it like a scale – if you add weight to one side, you must add the same weight to the other side to keep it balanced.

Example 3: Solve for x: x + 5 = 12 Identify the operation being done to x: x is being added to

5. Use the inverse operation (subtraction) to isolate x: Subtract 5 from both sides of the equation. x + 5 - 5 = 12 - 5 Simplify: x = 7 Verification: Substitute x=7 back into the original equation: 7 + 5 =

1

2. This is true, so our solution is correct.

Example 4: Solve for y: y - 3 = 8 Identify the operation being done to y: y is having 3 subtracted from it. Use the inverse operation (addition) to isolate y: Add 3 to both sides of the equation. y - 3 + 3 = 8 + 3 Simplify: y = 11 Verification: Substitute y=11 back into the original equation: 11 - 3 =

8. This is true, so our solution is correct.

Example 5: Solve for a: 2a = 10 Identify the operation being done to a: a is being multiplied by

2. Use the inverse operation (division) to isolate a: Divide both sides of the equation by 2. 2a / 2 = 10 / 2 Simplify: a = 5 Verification: Substitute a=5 back into the original equation: 2 * 5 =

1

0. This is true, so our solution is correct.

Example 6: Solve for b: b / 4 = 3 Identify the operation being done to b: b is being divided by

4. Use the inverse operation (multiplication) to isolate b: Multiply both sides of the equation by 4. (b / 4) 4 = 3 4 Simplify: b = 12 Verification: Substitute b=12 back into the original equation: 12 / 4 =

3. This is true, so our solution is correct. Guided Practice (With Solutions)

Question 1: Simplify the expression: 4x + 7 - 2x + 3 Solution: Identify like terms: 4x and -2x; 7 and

3. Combine like terms: (4x - 2x) + (7 + 3)

Simplify: 2x + 10

Commentary: We grouped the 'x' terms together and the constant terms together. Remember to pay attention to the signs (+ or -) in front of each term.

Question 2: Solve for m: m - 6 = 15 Solution: Identify the operation being done to m: 6 is being subtracted from m.

Use the inverse operation (addition): Add 6 to both sides of the equation. m - 6 + 6 = 15 + 6 Simplify: m = 21 Verification: Substitute m=21 back into the original equation: 21 - 6 =

1

5. This is true, so our solution is correct.

Commentary: To isolate 'm', we added 6 to both sides. Adding 6 'undoes' the subtraction of

6. Question 3: Solve for p: 3p = 21 Solution: Identify the operation being done to p: p is being multiplied by

3. Use the inverse operation (division): Divide both sides of the equation by 3. 3p / 3 = 21 / 3 Simplify: p = 7 Verification: Substitute p=7 back into the original equation: 3 * 7 =

2

1. This is true, so our solution is correct.

Commentary: To isolate 'p', we divided both sides by

3. Dividing by 3 'undoes' the multiplication by

3. Question 4: Simplify: 2y + 5x - y + 3 - 2x + 1 Solution: Identify like terms: 2y and -y; 5x and -2x; 3 and

1. Combine like terms: (2y - y) + (5x - 2x) + (3 + 1)

Simplify: y + 3x + 4

Commentary: Be careful to group the x and y terms correctly.