Solving Simultaneous Equations

What are Simultaneous Equations?

Imagine you're told that two numbers, x and y, add up to 10 (x + y = 10). There are infinite possibilities: x could be 1 and y could be 9, or x could be 5.5 and y could be 4.5, and so on.

Now, what if you're also told that the difference between these same two numbers is 4 (x - y = 4)? This second condition constrains the possibilities. There is now only one pair of values for x and y that is true for both equations simultaneously. Our goal is to find that unique solution.

We will focus on two key algebraic methods to do this: elimination and substitution.

Method 1: The Elimination Method

The elimination method aims to remove one of the variables by adding or subtracting the two equations. This works best when the equations are both in the form ax + by = c.

To start, let's watch a video that demonstrates the core mechanics of this method.

The Core Process of Elimination

As you saw, the process depends on the coefficients of the variables:

1. Label your equations (1) and (2). This is good practice for keeping your work clear.
2. Check the coefficients of x and y in both equations.
   • Case A (Opposite Coefficients): If the coefficients of one variable are opposites (e.g., +3y and -3y), you can add the two equations to eliminate that variable.
   • Case B (Same Coefficients): If the coefficients of one variable are the same (e.g., +5x and +5x), you can subtract one equation from the other to eliminate it.
   • Case C (Different Coefficients): If no coefficients match, multiply one or both equations by a number to create a match. Aim for the Lowest Common Multiple (LCM) of the coefficients. For example, to match 2y and 3y, you would multiply the first equation by 3 (to get 6y) and the second by 2 (to get 6y).
3. Solve the resulting single-variable equation.
4. Substitute the value you just found back into one of the original equations to find the second variable.
5. Check your answer by substituting both values (x and y) into the other original equation. This is a crucial step to catch any arithmetic errors, which you've noted can be an issue.

A handy rule to remember for step 2 is SAD-SAM:

• If the signs are the Same, you Subtract.
• If the signs are Different, you Add.

Let's review this process with a detailed text example.

Method 2: The Substitution Method

The substitution method involves rearranging one equation to make one variable the subject (e.g., y = ...) and then substituting this expression into the other equation.

This brings us to a key strategic point you've mentioned: choosing the most efficient method.

The Core Process of Substitution

1. Look for a variable with a coefficient of 1 or -1. Rearrange that equation to make that variable the subject. For example, rearrange 3x + y = 10 into y = 10 - 3x.
2. Substitute this expression into the other equation. This creates a new equation with only one variable.
3. Solve this new equation.
4. Substitute the result back into your rearranged equation from step 1 to find the second variable.
5. Check your answer using the original equation you didn't rearrange.

Let's see this in action.

Formulating and Solving from Word Problems

In IGCSE exams, you will often need to formulate the equations yourself from a written description before you can solve them. This is where your problem-solving skills come into play.

The process is:

1. Define your variables: State clearly what x and y (or any letters) represent. E.g., "Let c be the cost of a coffee and t be the cost of a tea."
2. Translate the information: Read the problem carefully and write down two distinct equations based on the information given.
3. Solve: Choose the most efficient method (elimination or substitution) to solve the system.
4. Answer in context: State your final answer clearly, making sure to include units (£, cm, etc.) and relate it back to the original question.

The following video works through several excellent examples of this, including one that directly relates to your weakness with "show that" questions in a geometric context.

By watching that geometry example, you can see that "show that" in this context simply means "use the properties of a rectangle (opposite sides are equal) to form an algebraic equation and simplify it to the required format". This directly tackles one of your identified weak points.

Conclusion

Today, we've added a vital tool to your algebraic toolkit. You now have two reliable methods for solving linear simultaneous equations and a strategy for choosing the most efficient one. More importantly, you've seen how to apply these methods to word problems, translating real-world scenarios into solvable mathematical systems.

Key Takeaways:

• Elimination Method: Best when both equations are in ax + by = c form. Make coefficients of one variable match, then add or subtract to eliminate it. Remember: SAD-SAM (Same signs -> Subtract, Different signs -> Add).
• Substitution Method: Most efficient when one variable has a coefficient of 1 or -1, allowing for easy rearrangement.
• Word Problems: Follow the four-step process: Define -> Translate -> Solve -> Answer in context.
• Always check your work! Substituting your final x and y values back into an original equation is the best way to guarantee accuracy and avoid losing marks to small errors.

Preview of the Next Lesson:
We will build directly on today's skills in the next lesson as we tackle a slightly more complex topic: solving simultaneous equations where one is linear and one is non-linear. This will require us to combine the substitution method with our knowledge of solving quadratic equations.