Mr Barton’s Maths Notes Algebra 4. Solving Linear Equations www.mrbartonmaths.com

4. Solving Linear Equations What on earth does Solving Equations mean? Let’s look at each of these three words in turn… Equations – these are just the same as expressions (what we have looked at in the last 3 sections, but with an equals sign (=) thrown in for good measure Linear – this just means we don’t have to worry about annoying powers… just yet! Solving – this means we must find the value of the unknown which makes the equation balance Now, there are a lot of different ways to solve equations, and if you are happy with the way that you have been taught, then stick to it, but this is the way I do them… How Mr Barton Solves Equations Golden Rule: Whatever you do to one side of the equation, you must do exactly the same to the other side to keep the equation in balance Aim: To be left with your unknown letter on one side of the equals sign, and a number on the other side Method: By doing the same to both sides of the equation… 1. If they are not already, get all your unknown letters on one side of the equation (NOT on the bottom of fractions and avoiding negatives). 2. Begin unwrapping your unknown letter, by thinking about the order that things were done to the letter 3. Use inverse operations to do this until you are left with just your unknown letter on one side, and the answer on the other 4. Check your answer using substitution and you should never ever get one of these wrong!

What are Inverse Operations?... Inverse operations are the key to solving equations as they allow you to unwrap all the things surrounding your unknown letter and leave you with a simple answer. Inverse operations are just operations which are the opposite of each other, and as such they cancel each other out. Here are the main ones you need to know…. power of 2 Now, the way I am going to set out these first few examples may seem very long and painful, but if you can do it this way for the simple ones, there is no reason why you can do the same for the nightmare stinker ones at the end…

Example 1 1. Right, here we go… now our unknown letter (p) only appears on the left hand side of the equation, there is no negative sign in front of it, and it is not on the bottom of a fraction, so that’s a good start! 2. Okay, what order were things done to p… 3. And so now we can unwrap, starting with the last operation, and doing the inverse (opposite) to both sides: Add three to both sides Notice how the +3 cancels out the -3! Divide both sides by 7 Notice how dividing by 7 cancels out the 7 multiplying the p 4. We have our answer, but it’s so easy to check if we are right, that we might as well do it. Just substitute p = 5 into the questions, and hope the equation balances… When p = 5…

Example 2 1. Okay, so let’s do our checks… our unknown letter (r) only appears on the left hand side of the equation, there is no negative sign in front of it, and it is not on the bottom of a fraction, so we are good to go… after we expand the brackets, of course… 2. Okay, what order were things done to r… Note: if this bit confused you, have another read of 1. Rules of Algebra 3. And so now we can unwrap, starting with the last operation, and doing the inverse (opposite) to both sides: Subtract twelve from both sides Notice how the -12 cancels out the +12! Divide both sides by 6 Notice how dividing by 6 cancels out the 6 on the top! 4. We have our answer, but it’s so easy to check if we are right, that we might as well do it. Just substitute r = 4 into the questions, and hope the equation balances… When r = 4…

Example 3 1. Okay, so our unknown letter (k) only appears on the left hand side of the equation, there is no negative sign in front of it, and it is not on the bottom of a fraction. Phew! 2. Okay, what order were things done to k… Note: just because k is not written first, doesn’t change the order in which things are done to k! Think: BODMAS! 3. And so now we can unwrap, starting with the last operation, and doing the inverse (opposite) to both sides: Subtract six from both sides Again, look at the cancelling out! Multiply both sides by 5 It all cancels out! We have our answer, but it’s so easy to check if we are right, that we might as well do it. Just substitute k = -35 into the questions, and hope the equation balances… When k = -35…

Example 4 1. Okay, so let’s do our checks… our unknown letter (m) only appears on the left hand side of the equation, it’s not on the bottom of a fraction, but wait… it’s got a negative sign in front of it! This is going to make life difficult, but we can sort it out by using inverse operations to cancel out the -3m… We just need to add 3m to both sides! And now we have an equation just like all the others! 2. Okay, what order were things done to m… 3. And so now we can unwrap, starting with the last operation, and doing the inverse (opposite) to both sides: Subtract six from both sides The 6s on the right hand side will cancel! Divide both sides by 3 Substitution to check our answer: When m = 6…

Example 5 1. Okay, we have trouble right away! All of the unknowns (y) are NOT on the same side. No problem, we just need a bit of inverse operations. Top Tip: Collect your letters on the side which starts off with the most letters… so the right hand side! So, we just need to subtract 7y from both sides! And now we have an equation just like all the others! Note: if this bit confused you, have another read of 1. Rules of Algebra 2. Okay, what order were things done to y… 3. And so now we can unwrap, starting with the last operation, and doing the inverse (opposite) to both sides: Add six to both sides The 6s on the right hand side will cancel! Divide both sides by 3 Substitution to check our answer balances!: When y = 3… Left hand side Right hand side

Example 6 1. Problem! Our unknown letter (g) is on the bottom of a fraction! The only way we are going to get that g off the bottom of the fraction is to realise that the 25 is being divided by g – 1 and use inverse operations… So, we just need to multiply both sides by (g – 1) And expand the brackets on the right hand side And now we have an equation just like all the others! 2. Okay, what order were things done to g… 3. And so now we can unwrap, starting with the last operation, and doing the inverse (opposite) to both sides: Add five to both sides The 5s on the right hand side will cancel! Divide both sides by 5 Substitution to check our answer is correct!: When g = 6…

Good luck with your revision!