Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Identify and combine like terms in expressions.
• Simplify expressions with addition, subtraction, and multiplication.
• Solve simple one-step equations.
• Solve simple two-step equations.
• Apply these skills to solve word problems.

Lesson summary

This week, we delve deeper into the world of algebraic expressions and simple equations. Last week, we introduced the basics of variables and constants. Now, we'll learn how to simplify algebraic expressions by combining like terms and how to solve simple equations using inverse operations. This skill is crucial because algebra is a fundamental building block for all higher-level mathematics, including trigonometry, calculus, and statistics. Think of it as learning the alphabet before you can read and write! Knowing how to manipulate algebraic expressions and solve equations is not just about passing exams.

Lesson notes

2.1 What are Like Terms? Like terms are terms in an algebraic expression that have the same variable(s) raised to the same power. Only the coefficients (the numbers in front of the variables) can be different.

Example: `3x` and `5x` are like terms. Both have the variable `x` raised to the power of 1 (which is usually not written).

Example: `2y²` and `-7y²` are like terms. Both have the variable `y` raised to the power of

2. Example: `4ab` and `ab` are like terms. Both have the variables `a` and `b` each raised to the power of

1. Non-

Example: `2x` and `2x²` are not like terms. One has `x` raised to the power of 1, and the other has `x` raised to the power of

2. Non-

Example: `3xy` and `3x` are not like terms. One has both `x` and `y`, while the other only has `x`. 2.2 Simplifying Algebraic Expressions by Combining Like Terms To simplify an algebraic expression, we combine like terms by adding or subtracting their coefficients.

Example 1: Simplify `2x + 5x - 3x` All terms have `x` raised to the power of 1, so they are like terms.

Add/subtract the coefficients: 2 + 5 - 3 = 4 Therefore, `2x + 5x - 3x = 4x` Example 2: Simplify `4y + 2 - y + 6` `4y` and `-y` are like terms. `2` and `6` are like terms (constants). Combine the `y` terms: 4y - y = 3y Combine the constant terms: 2 + 6 = 8 Therefore, `4y + 2 - y + 6 = 3y + 8` Example 3: Simplify `5ab + 2a - 3ab + a` `5ab` and `-3ab` are like terms. `2a` and `a` are like terms. Combine the `ab` terms: 5ab - 3ab = 2ab Combine the `a` terms: 2a + a = 3a Therefore, `5ab + 2a - 3ab + a = 2ab + 3a` 2.3 Solving Simple One-Step Equations An equation is a statement that two expressions are equal. 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. An inverse operation "undoes" another operation.

Addition and Subtraction: Addition and subtraction are inverse operations.

Multiplication and Division: Multiplication and division are inverse operations. To solve an equation, we perform the same operation on both sides of the equation to maintain equality.

Example 1: Solve `x + 5 = 12` To isolate `x`, we need to undo the "+ 5". The inverse operation is "- 5".

Subtract 5 from both sides: `x + 5 - 5 = 12 - 5` Simplify: `x = 7` Example 2: Solve `y - 3 = 8` To isolate `y`, we need to undo the "- 3". The inverse operation is "+ 3".

Add 3 to both sides: `y - 3 + 3 = 8 + 3` Simplify: `y = 11` Example 3: Solve `3z = 15` `3z` means "3 times z". To isolate `z`, we need to undo the multiplication by

3. The inverse operation is division by

3. Divide both sides by 3: `3z / 3 = 15 / 3` Simplify: `z = 5` Example 4: Solve `w / 2 = 4` `w / 2` means "w divided by 2". To isolate `w`, we need to undo the division by

2. The inverse operation is multiplication by

2. Multiply both sides by 2: `(w / 2) 2 = 4 * 2` Simplify: `w = 8` 2.4 Solving Simple Two-Step Equations Two-step equations require two operations to isolate the variable. We generally follow the reverse order of operations (PEMDAS/BODMAS) when solving for the variable.

Example 1: Solve `2x + 3 = 11` First, undo the addition of 3 by subtracting 3 from both sides: `2x + 3 - 3 = 11 - 3` Simplify: `2x = 8` Next, undo the multiplication by 2 by dividing both sides by 2: `2x / 2 = 8 / 2` Simplify: `x = 4` Example 2: Solve `y / 4 - 1 = 2` First, undo the subtraction of 1 by adding 1 to both sides: `y / 4 - 1 + 1 = 2 + 1` Simplify: `y / 4 = 3` Next, undo the division by 4 by multiplying both sides by 4: `(y / 4) 4 = 3 * 4` Simplify: `y = 12` Guided Practice (With Solutions)

Question 1: Simplify the expression: `7a - 3a + 2b + 5b - a` Solution: Identify like terms: `7a`, `-3a`, and `-a` are like terms. `2b` and `5b` are like terms. Combine the `a` terms: `7a - 3a - a = 3a` Combine the `b` terms: `2b + 5b = 7b` Therefore, the simplified expression is `3a + 7b`.

Question 2: Solve the equation: `x - 7 = 15` Solution: To isolate `x`, add 7 to both sides of the equation: `x - 7 + 7 = 15 + 7` Simplify: `x = 22` Question 3: Solve the equation: `4y + 2 = 14` Solution: First, subtract 2 from both sides: `4y + 2 - 2 = 14 - 2` Simplify: `4y = 12` Next, divide both sides by 4: `4y / 4 = 12 / 4` Simplify: `y = 3` Question 4: Simplify the expression `2p + 5q - p + 3 - 2q` Solution: Identify like terms: `2p` and `-p` are like terms. `5q` and `-2q` are like terms. `3` is a constant term.

Combine p terms: `2p - p = p` Combine q terms: `5q - 2q = 3q` Therefore, the simplified expression is `p + 3q + 3`. Independent Practice (Questions Only)

Simplify: `6b + 2c - 4b + c - b` Simplify: `8x - 3y + 2x + 5y - 4x` Solve: `m + 9 = 20` Solve: `n - 4 = 11` Solve: `5p = 35` Solve: `q / 3 = 6` Solve: `2r - 5 = 9` Solve: `s / 2 + 3 = 7` Simplify: 3(a + 2b) – a + b Solve: 4x + 7 = 3x + 10