Solving Simultaneous Equations: The DVD and CD Dilemma (GCSE Maths)

Simultaneous equations are a common topic in GCSE Maths, and they often appear in word problems that require you to translate real-world scenarios into algebraic equations. Let’s tackle a classic example involving DVDs and CDs.

The Problem:

David buys 2 DVDs and 2 CDs in a shop, and in total they cost £18. Ellie buys 3 DVDs and 2 CDs in the same shop, and they cost £22. Find the cost of each DVD and each CD.

Step-by-Step Solution

1. Define Variables: Let’s use variables to represent the unknowns:
   • Let ‘d’ be the cost of one DVD (in pounds).
   • Let ‘c’ be the cost of one CD (in pounds).
2. Formulate Equations: Translate the given information into two equations:
   • Equation 1 (David): 2d + 2c = 18
   • Equation 2 (Ellie): 3d + 2c = 22
3. Solve the Equations: We can use several methods to solve simultaneous equations. In this case, the elimination method is particularly straightforward. Notice that the ‘c’ terms in both equations have the same coefficient (2). This means we can eliminate ‘c’ by subtracting one equation from the other.Subtract Equation 1 from Equation 2:(3d + 2c) – (2d + 2c) = 22 – 18This simplifies to:d = 4So, the cost of one DVD is £4.
4. Substitute to Find the Other Variable: Now that we know d = 4, we can substitute this value into either Equation 1 or Equation 2 to find the value of ‘c’. Let’s use Equation 1:2(4) + 2c = 188 + 2c = 182c = 10c = 5So, the cost of one CD is £5.

Answer:

• The cost of each DVD is £4.
• The cost of each CD is £5.

Checking Your Answer (Crucial!)

Always check your answer by substituting the values of ‘d’ and ‘c’ back into the original equations:

• Equation 1: 2(4) + 2(5) = 8 + 10 = 18 (Correct)
• Equation 2: 3(4) + 2(5) = 12 + 10 = 22 (Correct)

Since both equations are satisfied, our solution is correct.

Why Are Simultaneous Equations Important in GCSE Maths?

• They test your algebraic skills.
• They require you to translate word problems into mathematical equations.
• They develop your problem-solving abilities.
• They often appear on GCSE Maths exams.

Other Methods for Solving Simultaneous Equations

While we used the elimination method in this example, you can also use the substitution method or graphical methods to solve simultaneous equations.

Common Mistakes to Avoid

• Incorrectly setting up the equations. Make sure you accurately translate the information from the word problem into algebraic equations.
• Making arithmetic errors. Double-check your calculations to avoid mistakes.
• Forgetting to check your answer. Always substitute your solution back into the original equations to verify its correctness.

Need Extra Help with Simultaneous Equations?

If you find simultaneous equations challenging, don’t hesitate to seek help. GCSE Mathematics Tuition Online can provide 1-1 instruction and support to help you master this important topic. An online tutor can explain the concepts clearly, work through examples step-by-step, and provide you with targeted practice to build your confidence. They can also help you develop effective problem-solving strategies.

Solving simultaneous equations is a key skill for GCSE Maths. By understanding the concepts and practicing these types of problems, you can improve your algebraic skills and achieve success in your exams. Keep practicing, and remember to check your answers! Good luck!