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Homework Sheet – Linear Equations and Simultaneous Equations 1. Solve the following equations: (a) 3x + 5 = 2 (b) 4x - 5 = 2 (e) 2x + 3 = -6 (f) (i) 2(2x + 3) = 3(4x – 2) (j) x 4 =2 3 x x +4= +1 2 4 2 x = 12 3 (c) 4x = 50 (d) (g) 2(x + 3) = 10 (h) 3(x – 2) = 15 (k) x 2 x 1 – =2 3 4 2. A number is multiplied by 5, and then 9 is subtracted. This gives an answer of 26. Find the number. 3. The sum of three consecutive numbers is 219. Let the first number be x. Form and solve an equation in x to find the numbers. 4. The sum of four consecutive odd numbers is 376. Let the first number be x. What is the largest of the four numbers? 5. Use a graphical method to solve each of these pairs of simultaneous equations: (a) x + y = 6 y=x–2 6. Solve each of these pairs of simultaneous equations, making your method clear: (a) 3x + 2y = 8 2x – y = 3 7. (b) 3x + 4y = 24 2y = x + 2 (b) 5x + y = 16 y = 3x (c) x + 4y = 32 x = 2y – 4 (d) x2 + y2 = 20 x = 2y A straight line has the equation y = 5x + 3 A curve has the equation y = x2 + 2x + 3 Solve these simultaneous equations to find the points of intersection of the line and the curve. Linear Equations and Simultaneous Equations Q 1(a) 1(b) 1(c) Answer 1(e) 1(g) 1(h) 1(i) Comments M1 x = -1 A1 4x = 2 + 5 M1 x=7÷4 Alternative methods (e.g. flow diagrams) are acceptable. M1 or 1.75 3 x=1 4 A1 x = 50 ÷ 4 M1 1 2 A1 x = 12 × 3 ÷ 2 M1 x = 18 A1 2x = -6 – 3 M1 x = -9 ÷2 M1 1 x= 4 2 1(f) Mark 3x = 2 – 3 x = 12 1(d) Answers and Mark Scheme Students should be encouraged to show their working. or 12.5 delete sentences or -4.5 A1 x =-2+4=2 3 M1 x=2x3 M1 x=6 A1 x + 3 = 10 ÷ 2 = 5 M1 x=5–3 M1 x=2 A1 x - 2 = 15 ÷ 3 = 5 M1 or 3x – 6 = 15 x=5+2 M1 3x = 15 + 6 x=7 A1 4x + 6 = 12x – 6 M1 Multiply out brackets 8x = 12 M1 Gather like terms x = 12 ÷ 8 M1 x=1 1 2 A1 or 2x + 6 = 10 2x = 10 – 6 or 1.5 Q 1(j) 1(k) 2. 3. 4. 5(a) Answer Mark Comments 2x + 16 = x + 4 M1 Multiply throughout by 4 2x – x = 4 – 16 M1 Gather like terms x = -12 A1 4(x + 2) – 3(x- 1) = 24 M1 Multiply throughout by 12 (4x + 8) – (3x – 3) = 24 M1 Multiply out brackets x +11 = 24 M1 Gather like terms x = 24 – 11 M1 x = 13 A1 5x – 9 = 26 M1 5x = 26 + 9 = 35 M1 x = 35 ÷ 5 = 7 M1 The number is 7 A1 x + (x + 1) + (x + 2) = 219 M1 3x + 3 = 219 M1 3x = 219 – 3 = 216 M1 x = 216 ÷ 3 = 72 A1 The numbers are 72, 73 & 74 A1 x + (x + 2) + (x+ 4) + (x+ 6) = 376 M1 4x + 12 = 376 M1 4x = 376 – 12 = 364 M1 x = 364 ÷ 4 M1 x = 91 A1 The largest of the four numbers is 97 A1 Set of axes, at least 0≤x≤6, 0≤y≤6, fully labelled B1 x + y = 6 drawn correctly M1 y = x – 2 drawn correctly M1 x=4 &y=2 A1 A1 Form equation Students should be encouraged to give answers in words to questions in words. Students should be encouraged to give answers in words to questions in words. Q 5(b) Answer Mark Set of axes, at least 0≤x≤8, 0≤y≤6, fully labelled B1 3x + 4y = 24 drawn correctly M1 2y = x + 2 M1 drawn correctly x=4 &y=3 6(a) A1 A1 3x + 2y = 8 (x2) -> 6x +4y = 16 2x – y = 3 ( x3) -> 6x – 3y = 9 Subtract 6(b) 6(c) 7y = 7 Selecting appropriate multiples of equations M1 A1 -> x = 2 A1 Subtracting M1 Appropriate substitution 5x + y = 16 y = 3x -> 5x + 3x = 16 -> 8x = 16 M2 -> x = 2 A1 -> y = 6 A1 M1 Solving linear equation M1 Appropriate substitution x + 4y = 32 M2 -> y=6 A1 -> x=8 A1 x2+ y2 = 20 M1 Solving linear equation M1 Appropriate substitution M1 Solving linear equation -> 4y2 + y2 = 20 -> 5y2 = 20 x = 2y -> y2= 4 -> y = +2 or y = -2 -> x = +4 or x = -4 i.e. x = 4 & y = 2 7. M1 -> y = 1 x = 2y – 4 -> 2y – 4 + 4y = 32 -> 6y = 36 6(d) Comments or x = -4 & y = -2 M2 A2 A2 x2 + 2x + 3 = 5x + 3 M1 x – 3x = 0 M1 x(x – 3) = 0 M1 x = 0 or x = 3 A1 points of intersection are (0, 3) and (3, 18) A1 2 Solutions must be correctly linked