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2. EQUATIONS The important thing to remember about equations is that both sides must balance (both sides must equal each other). This means that if you do something to one side of the equation you must also do the same to the other: e.g. 5 + 2 = 3 + 4 If you take 2 away from the first part of the equation e.g. 5 + 2 = 3 + 4 YOU MUST ALSO take 2 away from the second part e.g. 5 + 2 = 3 + 2 Click mouse So you can see, the equation is still balanced! e.g. 5 + 2 = 3 + 4 77 55

3. Look at this example, and remember the balance! Remember, if you do something to one side of the equation you must also do the same to the other: e.g. 6 + 4 = 3 + 7 If you take 4 away from the first part of the equation e.g. 6 + 2 = 3 + 7 YOU MUST ALSO take 4 away from the second part e.g. 6 + 2 = 3 + 3 Click mouse So you can see, the equation is still balanced! e.g. 6 + 4 = 3 + 7 10 66

4. You can do anything to the equation, IF you do the same to both sides. So, if you ADD something to one side of the equation you must also ADD the same to the other: e.g. 6 + 4 = 3 + 7 If you add 4 to the first part of the equation e.g. 6+ 4 + 4 = 3 + 7 YOU MUST ALSO add 4 to the second part e.g. 6 + 4 + 4 = 3 + 7 + 4 Click mouse So you can see, the equation is still balanced! e.g. 6 + 4 = 3 + 7 10 14

5. You can do anything to the equation, IF you do the same to both sides. So, if you DIVIDE one side of the equation by a number you must also DIVIDE the other side by the same: e.g. 6 + 4 = 3 + 7 If you DIVIDE the first part of the equation by 2 e.g. 6 + 4 = 3 + 7 2 YOU MUST ALSO DIVIDE the second part by 2 e.g. 6 + 4 = 3 + 7 2 2 Click mouse So you can see, the equation is still balanced! e.g. 6 + 4 = 3 + 7 10 55

6. You can do anything to the equation, IF you do the same to both sides. So, if you MULTIPY one side of the equation by a number you must also MULTIPLY the other side by the same: e.g. 6 + 4 = 3 + 7 If you MULTIPLY the first part of the equation by 2 e.g. (6 + 4) x 2 = 3 + 7 YOU MUST ALSO MULTIPLY the second part by 2 e.g. (6 + 4) x 2 = (3 + 7) x2 Click mouse So you can see, the equation is still balanced! e.g. 6 + 4 = 3 + 7 10 20

7. Now that you know these facts they will really help you to solve equations. RULE: Equations must always balance. Click mouse EQUATIONS Whatever you do to one side you must ALSO do to the other.

8. BALANCE EQUATIONS MUST ALWAYS BALANCE. This fact will help you to answer the following questions. Have a go at them. 1. ….. + 60 = 100 What must you add to 60 to make 100 balanced, The equation is now balanced, because both sides equal 100 Click mouse 40 1. ….. - 7 = 3 What number take away 7 makes 3 balanced, The equation is now balanced, because both sides equal 3 10

9. Both those sums were very easy and you could work them out in your head. However, if you understand what you are doing it will help you to work out harder sums!! Let’s look again at the first one we did and think about how we did it:- …… + 60 = 100 What we really did was to take 60 away from BOTH SIDES This left us with the answer we needed! …… + 60 = 100…… + 60 = 100 - 60…… + 60 = 40 Take away 60 Click mouse

10. Let’s look again at the second one we did and think about how we did it:- ….. - 7 = 3 What we really did was to ADD 7 to BOTH SIDES This left us with the answer we needed! Add 7 Click mouse It’s important to remember that this is a MINUS 7 Add 7 ( to get rid of the –7 ) ….. - 7 = 3…… + 60= 3…… + 60 = 3 + 7…… + 60 = 10

11. Let’s try adding a little algebra! Instead of leaving the gap … we put a letter into the equation. Look at this equation and work it out:- y + 40 = 60 What we must do is take 40 away from BOTH SIDES to leave the y by itself This left us with the answer we needed! y = 20 y + 60 = 60 y + 60 = 60 - 40 y + 60 = 20 Take away 40 Click mouse Find the value of y

12. Now try these – they’re easy!!!! z + 3 = 15 Take 3 away from BOTH SIDES to leave the z by itself z + 60 = 15 z + 60 = 15 - 3 z + 60 = 12 Take away 3 Click mouse Find the value of z y + 20 = 35 Take 20 away from BOTH SIDES to leave the y by itself y + 60 = 35 y + 60 = 35 - 20 y + 60 = 15 Take away 20 Find the value of y

13. Now try these with MINUS numbers in them – they’re still easy if you remember the rule!! z - 3 = 15 Add 3 to BOTH SIDES to leave the z by itself z + 60 = 15 z + 60 = 15 + 3 z + 60 = 18 ADD 3 ADD 3 to get rid of the -3 Click mouse Find the value of z y - 20 = 35 Add 20 to BOTH SIDES to leave the y by itself y + 60 = 35 y + 60 = 35 + 20 y + 60 = 55 Add 20 ADD 20 to get rid of the -20 Find the value of y Remember, this is a MINUS

14. Can you see the pattern? KEEPING THE BALANCE What you are doing is KEEPING THE BALANCE If it is a PLUS on one side of the equation, it becomes a MINUS on the other. This is called INVERSING If it is a MINUS on one side of the equation, it becomes a PLUS on the other. 2 + 3 = 5 + - 2 + 3 = 5 - 3 SO…….. 2 = 2 8 - 3 = 5 - + 8 + 3 = 5 + 3 SO…….. 8 = 8 Click Mouse

15. Try these. Click the mouse for answer / help Find the value of x x + 8 = 21 + - x + 8 = 21 x + 8 = 21 - 8 x + 8 = 13 Find the value of y y - 8 = 22 - + y - 8 = 22 y = 22 + 8 y + 8 = 30 Find the value of z z + 19 = 21 + - z + 19 = 21z = 21 - 19z = 2 Find the value of x x - 19 = 5 - + x - 19 = 5x = 5 + 19x = 24 Find the value of b b - 5 = 26 - + b - 5 = 26b = 26 + 5b = 31 Find the value of a a + 7 = 21 + - a + 7 = 21a = 21 - 7a = 14 Click mouse

16. Now you’ve got the hang of these, let’s try with multiplication and division The same rules apply – you must keep your equation balanced Look at this equation:- 4 x 3 = 12 This is balanced (because 4x3 is 12) To get rid of the x3 we must divide by 3, ON BOTH SIDES OF THE EQUATION 4 x 3 = 12 Divide by 3 to get rid of x3 4 x 3 = 124 x 3 = 12 ÷ 34 x 3 = 4 So divide this side by 3 Balanced again! Click mouse Not yet balanced!!!

17. Keep your equation balanced Look at this equation:- 4 ÷ 2 = 2 This is balanced (because 4 ÷ 2 is 2) To get rid of the ÷ 2 we must multiply by 2, ON BOTH SIDES OF THE EQUATION 4 ÷ 2 = 2 Multiply by 2 to get rid of ÷ 2 4 x 3 = 24 x 3 = 2 x 24 x 3 = 4 So multiply this side by 2 Balanced again! Click mouse Not yet balanced!!!

18. So, now that you can see the same rules apply, let’s try it with algebra Look at this equation:- y ÷ 2 = 8 To get rid of the ÷ 2 we must multiply by 2, ON BOTH SIDES OF THE EQUATION y ÷ 2 = 8 Multiply by 2 to get rid of ÷ 2 y x 3 = 8y x 3 = 8 x 2y x 3 = 16 So multiply this side by 2 Balanced again! Click mouse Not yet balanced!!!

19. Look at this equation:- y x 2 = 8 To get rid of the x 2 we must DIVIDE by 2, ON BOTH SIDES OF THE EQUATION y x 2 = 8 Divide by 2 to get rid of x 2 y x 3 = 8y x 3 = 8 ÷ 2y x 3 = 4 So divide this side by 2 Balanced again! Click mouse Not yet balanced!!!

20. Can you see the pattern? KEEPING THE BALANCE What you are doing is KEEPING THE BALANCE If it is a MULTIPLY on one side of the equation, it becomes a DIVIDE on the other - INVERSING again! If it is a DIVIDE on one side of the equation, it becomes a MULTIPLY on the other. 2 x 3 = 6 x ÷ 2 + 3 = 6 ÷ 3 SO…….. 2 = 2 8 ÷ 2 = 4 ÷ x 8 + 3 = 4 x 2 SO…….. 8 = 8 Click Mouse

21. Try these. Click the mouse for answer / help Find the value of b b x 8 = 24 x ÷ b x 8 = 24 b + 8 = 24 ÷ 8 b + 8 = 3 Find the value of y y ÷ 4 = 9 ÷ x y ÷ 4 = 9Y = 9 x 4 y + 8 = 36 Find the value of z z ÷ 8 = 6 ÷ x z ÷ 8 = 6z = 6 x 8z = 48 Find the value of a a x 7 = 49 x ÷ a x 7 = 49a = 49 ÷ 7a = 7 Find the value of b b x 5 = 40 x ÷ b x 5 = 40b = 40 ÷ 5b = 8 Find the value of a a ÷ 3 = 7 ÷ x a ÷ 3 = 7a = 7 x 3 a = 21 Click mouse