Understanding Simultaneous Equations

Getting Started with Simultaneous Equations

When you see two equations with the same variables, you're dealing with simultaneous equations. The goal is to find the values of both x and y that make both equations true at the same time.

Your first step is always to label your equations - this keeps you organised and prevents silly mistakes. Simply write ① next to the first equation and ② next to the second one.

Quick Tip: Always label your equations from the start - it'll save you loads of confusion later!

Making the Coefficients Match

Now you need to make one of the variables have the same coefficient in both equations. Look at the numbers in front of y (or x) and find their lowest common multiple.

In this example, 3 and 2 both go into 6. Multiply the first equation by 2 to get -6y, and multiply the second equation by 3 to get +6y. Remember to multiply the entire equation, not just the y term!

This gives you two new equations with matching y coefficients. Label these as equations ③ and ④ to stay organised.

Remember: Whatever you do to one side of an equation, you must do to the other side too!

Eliminating Variables

Here's where the magic happens! Look at the signs in front of your matching coefficients. If they're opposite signs (one positive, one negative), you add the equations together. If they're the same signs, you subtract.

Since -6y and +6y are opposites, add the equations. The y terms cancel out completely, leaving you with just x terms. Solve this simple equation to find your x value.

In this case, 17x = -34, so x = -2. You've successfully found half the answer!

Key Point: Opposite signs mean add, same signs mean subtract - this always eliminates one variable!

Finding the Second Variable

Time for substitution! Take your x value and plug it back into either of your original equations - pick whichever looks easier to work with.

Replace x with -2 in the equation 5x + 2y = 4. This becomes 5$-2$ + 2y = 4, which simplifies to -10 + 2y = 4. Rearrange to get 2y = 14, so y = 7.

You now have both values: x = -2 and y = 7. But don't celebrate just yet - there's one crucial step left!

Pro Tip: Choose the simpler-looking original equation for substitution - it'll make your calculations much easier!

Checking Your Answer

Never skip the checking step - it catches any calculation errors and confirms you've got the right answer. Use the same equation you picked for substitution.

Substitute both x = -2 and y = 7 into 5x + 2y = 4. Work it out: 5$-2$ + 2(7) = -10 + 14 = 4. Perfect! This matches the right-hand side of your equation.

If your check doesn't work out, go back through your steps to find where you made an error. Getting into this habit will boost your confidence and improve your marks.

Final Check: Your substituted values should make the original equation true - if they don't, retrace your steps!