# Mr Barton’s Maths Notes

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Mr Barton’s Maths Notes
Algebra 8. Simultaneous Equations

8. Simultaneous Equations
What are Simultaneous Equations? Simultaneous Equations are two equations, each containing two unknown letters, and you have to use both equations, in a clever way, to find the value of your unknown letters! Key Point: The values you find for your unknown letters must make BOTH equations balance – and once again this is another Algebra topic where you can check your answers and guarantee that you have got it right! I told you Algebra wasn’t so bad… Skills you need to have mastered before we start… In this section I am going to assume that you are an world expert on the following things: How to solve equations (see Algebra 4. Solving Equations) Rules of Algebra (see Algebra 1. Rules of Algebra) Rules of Negative Numbers (see Number 8. Negative Numbers) If this is not the case, go back now and have a quick read through! Please Note: The graphical method for solving simultaneous equations is discussed in Graphs 1. Straight Line Graphs

How Mr Barton Solves Simultaneous Equations

Example 1 1. Good news! Our equations are in the same form: some x’s and some y’s, equal a number! 1 2. Let’s write the second equation underneath the first… 2 3. Okay, so we need to pick either the x’s or the y’s to be our Key Letter. Well… notice how there are already the same number of y’s in both equations (there is a disguised 1 in front of both), so let’s pick the y’s to make life easier for ourselves! 1 4. Put a box around our Key Letters, and their signs: 2 5. The signs of our Key Letters are the same (both +) so we must Subtract equation from equation . 2 1 6. Our Key Letters have cancelled out, leaving us with a nice looking equation: ÷ 2 7. Solve it: 8. Use this value in one of the original equations (I’ll chose ) to find the value of the other unknown letter: 1 1 x = 5 9. And now we have our two answers: But we may as well check them using equation - 15 2 x = 5 y = 4 2

Example 2 1. Good news! Our equations are in the same form: some x’s and some y’s, equal a number! 1 2. Let’s write the second equation underneath the first… 2 3. Okay, so we need to pick either the x’s or the y’s to be our Key Letter. Well… notice how there are already the same number of y’s in both equations (there 2 – don’t worry about the sign!), so let’s pick the y’s to make life easier for ourselves! 1 4. Put a box around our Key Letters, and their signs: + 2 5. The signs of our Key Letters are different (- and +) so we must Add equation to equation . 2 1 6. Our Key Letters have cancelled out, leaving us with a nice looking equation: ÷ 5 7. Solve it: 8. Use this value in one of the original equations (I’ll chose ) to find the value of the other unknown letter: 2 2 x = 3 9. And now we have our two answers: But we may as well check them using equation - 6 1 ÷ 2 1 x = 3 y = 3

Example 3 1 1. Good news! Our equations are in the same form: some x’s and some y’s, equal a number! 2 2. Let’s write the second equation underneath the first… 1 x 3 3. Okay, bad news. We don’t have the same number of either unknown. No problem, though! Why not make the number of x’s the same by… multiplying by 3 and… multiplying by Note: We could have made the y’s the same if we had liked! 2 x 2 1 2 1 4. Put a box around our Key Letters, and their signs: 2 5. The signs of our Key Letters are the same (disguised +) so we must Subtract equation from equation . 2 1 6. Our Key Letters have cancelled out, leaving us with a nice looking equation: ÷ -1 7. Solve it (people seem to mess these ones up…) 8. Use this value in one of the original equations (I’ll chose ) to find the value of the other unknown letter: 2 2 y = -15 9. And now we have our two answers: But we may as well check them using equation - 75 1 ÷ 3 1 x = -19 y = 15

Example 4 1 1. Bad news! Look at that 2nd equation! Might just have to add 4y to both sides to sort that mess out! 2 2. Let’s write the second equation underneath the first… 1 x 2 3. Okay, bad news. We don’t have the same number of either unknown. No problem, though! Why not make the number of y’s the same by… multiplying by 2. The signs will be different, but who cares? 1 1 4. Put a box around our Key Letters, and their signs: + 2 5. The signs of our Key Letters are different (- and +) so we must Add equation to equation . 2 1 6. Our Key Letters have cancelled out, leaving us with a nice looking equation: ÷ 17 7. Solve it (be careful with negatives!) 8. Use this value in one of the original equations (I’ll chose ) to find the value of the other unknown letter: 2 2 x = -2 9. And now we have our two answers: But we may as well check them using equation + 6 1 ÷ 4 1 x = -2 y = 3