Let's dive into Grade 11 Maths, specifically tackling Unit 2 Exercise 26. This guide will break down each problem, offering clear, step-by-step solutions to help you master the concepts. Whether you're struggling with algebra or need a quick refresher, we've got you covered. So, grab your notebooks, and let's get started!

Understanding the Basics

Before we jump into the exercise, it's essential to grasp the foundational concepts covered in Unit 2. This unit typically revolves around algebraic expressions, equations, and inequalities. A solid understanding of these basics is crucial for solving the problems in Exercise 26 effectively. Remember, practice makes perfect, so don't hesitate to revisit earlier sections if you feel shaky on any particular topic.

Algebraic Expressions

Algebraic expressions are mathematical phrases that combine numbers, variables, and operation symbols (like +, -, ×, ÷). Simplifying these expressions often involves combining like terms and applying the distributive property. For example, consider the expression 3x + 2y - x + 5y. To simplify it, you would combine the x terms (3x - x = 2x) and the y terms (2y + 5y = 7y), resulting in the simplified expression 2x + 7y. Understanding how to manipulate these expressions is fundamental for more complex problem-solving.

Equations

Equations are mathematical statements that assert the equality of two expressions. Solving an equation means finding the value(s) of the variable(s) that make the equation true. For instance, in the equation 2x + 5 = 11, you would subtract 5 from both sides to isolate the term with x, giving 2x = 6. Then, divide both sides by 2 to find x = 3. Mastering equation-solving techniques is vital for tackling various mathematical problems.

Inequalities

Inequalities are mathematical statements that compare two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Solving inequalities involves similar steps to solving equations, but with one crucial difference: when multiplying or dividing both sides by a negative number, you must reverse the inequality sign. For example, if you have the inequality -3x < 9, dividing both sides by -3 would give x > -3. Understanding these nuances is key to accurately solving inequalities.

Exercise 26: Problem Breakdown and Solutions

Now, let's break down Exercise 26. We'll go through each problem step-by-step, providing detailed explanations and solutions. Remember, the goal is not just to get the right answer but to understand the process behind it. So, let's roll up our sleeves and get to work!

Question 1: Simplifying Algebraic Expressions

Problem: Simplify the expression: 5(a + 2b) - 3(2a - b)

Solution:

1. Apply the distributive property:
   • 5(a + 2b) = 5a + 10b
   • -3(2a - b) = -6a + 3b
2. Combine like terms:
   • (5a - 6a) + (10b + 3b)
3. Simplify:
   • -a + 13b

Answer: The simplified expression is -a + 13b. This problem emphasizes the importance of correctly applying the distributive property and combining like terms. Make sure you double-check your signs to avoid common errors.

Question 2: Solving Linear Equations

Problem: Solve for x: 4x - 7 = 9

Solution:

1. Add 7 to both sides:
   • 4x - 7 + 7 = 9 + 7
   • 4x = 16
2. Divide both sides by 4:
   • 4x / 4 = 16 / 4
   • x = 4

Answer: x = 4. This is a straightforward linear equation. The key is to isolate x by performing the same operations on both sides of the equation. Always verify your solution by plugging it back into the original equation.

Question 3: Solving Linear Inequalities

Problem: Solve for y: 3y + 5 < 14

Solution:

1. Subtract 5 from both sides:
   • 3y + 5 - 5 < 14 - 5
   • 3y < 9
2. Divide both sides by 3:
   • 3y / 3 < 9 / 3
   • y < 3

Answer: y < 3. Remember that since we didn't multiply or divide by a negative number, the inequality sign remains the same. The solution represents all values of y that are less than 3.

Question 4: Working with Systems of Equations

Problem: Solve the following system of equations:

• x + y = 5
• 2x - y = 1

Solution:

1. Use the elimination method: Add the two equations together.
   • (x + y) + (2x - y) = 5 + 1
   • 3x = 6
2. Solve for x:
   • x = 6 / 3
   • x = 2
3. Substitute the value of x into one of the original equations to solve for y:
   • 2 + y = 5
   • y = 5 - 2
   • y = 3

Answer: x = 2, y = 3. This problem demonstrates how to solve a system of equations using the elimination method. Alternatively, you could use the substitution method to arrive at the same solution.

Question 5: Applying Algebraic Concepts to Word Problems

Problem: A rectangle has a length that is twice its width. If the perimeter of the rectangle is 36 cm, find the length and width.

Solution:

1. Define variables:
   • Let w be the width of the rectangle.
   • Then, the length l = 2w.
2. Write the equation for the perimeter:
   • Perimeter = 2l + 2w
   • 36 = 2(2w) + 2w
3. Simplify and solve for w:
   • 36 = 4w + 2w
   • 36 = 6w
   • w = 36 / 6
   • w = 6
4. Find the length:
   • l = 2w = 2(6) = 12

Answer: The width of the rectangle is 6 cm, and the length is 12 cm. Word problems require careful reading and translating the given information into algebraic equations. Always double-check that your answer makes sense in the context of the problem.

Tips for Success

To excel in maths, especially in Grade 11, here are some tips to keep in mind:

• Practice Regularly: Maths is a skill that improves with consistent practice. Set aside time each day to work on problems and review concepts.
• Understand the Concepts: Don't just memorize formulas; understand the underlying concepts. This will help you apply your knowledge to a wider range of problems.
• Seek Help When Needed: Don't hesitate to ask your teacher, classmates, or online resources for help if you're struggling with a particular topic.
• Review Your Mistakes: When you make a mistake, take the time to understand why you made it and how to avoid it in the future.
• Stay Organized: Keep your notes and assignments organized so you can easily find what you need when you're studying or working on problems.

Conclusion

Mastering Grade 11 Maths, including exercises like Unit 2 Exercise 26, requires a combination of understanding basic concepts, consistent practice, and effective problem-solving strategies. By breaking down each problem into manageable steps and focusing on understanding the underlying principles, you can build confidence and achieve success in maths. Remember, everyone learns at their own pace, so be patient with yourself and celebrate your progress along the way. Keep practicing, keep asking questions, and you'll be well on your way to mastering Grade 11 Maths!