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Introduction To Equation Solving. x x Solve : 2x + 1 = 4 Solving A Basic Equation. Consider the equation below: This can be represented by the scales and marbles below. The problem is to find out how many marbles are in the box to make the scales balance. x + 2 = 5 (-2) x (-2) = 3 The solution to the equation is x = 3 . That is to say that there are three marbles inside the box to keep the scales balanced. x Remove two marbles from each side. Further Example. Solve the equation : m + 4 = 7 (-4) Draw scales to represent the equation: m (-4) m = 3 There are three marbles in the box to keep the scales balanced. Hence : m=3 How many marbles can we remove from both sides? 4 marbles What Goes In The Box ? Solve the following equations: (1) x + 8 = 14 x=6 (2) t + 10 = 18 (5) f + 14 = 34 f = 20 (6) b + 13 = 32 t=8 b = 19 (3) w + 3 = 12 w=9 (7) g + 70 = 110 g = 40 (4) h + 7 = 20 h = 13 (8) s + 27 = 74 s = 47 More Than One Box. Draw scales to represent the equation: Solve the equation: 3x = 9 3 x x x x 3 = 3 There are three marbles in each box. (1) If there are 9 marbles in three boxes how many are in 1 box ? 3 marbles What Goes In The Box ? 2 Solve the following equations: (1) 2f = 10 f=5 (2) 3m = 12 m=4 (3) 4t = 12 t=3 (4) 6k = 24 k=4 (5) 5y = 45 y=9 (6) 9 d = 54 d=6 Further Example. Solve the equation: 2x + 4 = 12 (-4) Draw scales to represent the equation. x x (-4) 2x = 8 2 2 How many marbles can be removed from both sides? 4 4 marbles x = There are 4 marbles in each box. Hence x = 4 is the solution. If two boxes contain 8 marbles how many marbles are in one box? 4 marbles Solve the equation : 3x + 5 = 17 (-5) 3x x x x = 12 3 x (-5) Draw scales to represent the equation: 3 = 4 As there are 4 marbles in each box the solution is x=4 How many marbles can you remove from both sides? 5 marbles. If three boxes contain 12 marbles then how many marbles are in one box ? 4 marbles What Goes In The Box ? 3 Solve the equations below: (1) 2h + 6 = 20 (5) 4r + 14 = 26 h=7 r=3 (2) 3n + 11 = 26 (6) 8f + 21 = 37 n=5 (3) 4w + 5 = 29 f=2 w=6 (7) 6a + 32 = 44 a=2 (4) 7t + 3 = 31 (8) 9p + 32 = 59 t=4 p=3 Solving Equations With Subtractions. Solve the equation: 2y – 3 = 7 +3 y y +3 2y = 10 2 y Draw scales to represent the equation: 2 = 5 There are 5 marbles in each box. y=5 The take away three means that three marbles have fallen out of a box. If you place the three fallen marbles back in the box what must you do to the other side of the balance ? Add three marbles. Now solve as before. Further Example. Solve the equation: 3m – 7 = 11 +7 m m m +7 3m = 18 3 3 m Draw scales to represent the equation: = 6 There are 6 marbles in each box. m=6 How many marbles must be added to each side? 7 marbles. Now solve as before. What Goes In The Box ? 4. Solve the equations below: (1) 3n – 5 = 1 (5) 6d – 7 = 11 n=2 d=3 (2) 4f – 8 = 12 (6) 7r – 14 = 21 f=5 r=5 (3) 5h – 17 = 13 h=6 (7) 8v – 21 = 35 v=7 (4) 2k – 21 = 31 (8) 9p – 26 = 28 k = 26 p=6 More Complex Equations. Draw scales to show the equation: Solve the equation: 4m – 2 = m + 7 (+2) m m m m m (+2) 4m = m + 9 (- m) (- m) How many marbles can you put back ? 2 marbles. 3m = 9 How many boxes can you remove ? 3 3 1 box. m = 3 There are three marbles in each box . Further Example. Solve the equation: Draw scales to show the equation: 6g – 3 = 2g + 9 +3 g +3 6g g g g g g g g = 2g + 12 (-2g) (-2g) 4g = 12 4 4 How many marbles can we add to both sides ? 3 marbles. How many boxes can you remove from both sides? g = 3 2 boxes There are 3 marbles in each box. What Goes In The Box 5 ? Solve the equations below : (1) 4 m – 3 = m + 15 m=6 (2) 6y – 2 = 2y + 18 y=5 (3) 6t + 3 = 3t + 15 t=4 (4) 8r – 7 = 3r + 23 r=6 (5) 10 w – 9 = 3w + 33 w=6 (6) 9f + 14 = 3f + 56 f=7