32 a If one of the equations is a multiple of B1 2 B1 for clear explanation M the other then there will be an infinite number of solutions, e.g. B1 B1 for use of a good example accompanying the x y explanation + = 5 2x + 2y = 10 x y Every point on the line + = 5 is a solution, giving an infinite number of solutions. B1 B1 for clear explanation If the equations have graphs that are B1 B1 for use of a good example accompanying the parallel to each other then there will be explanation no intersection and so no solution, e.g. x y + = 5 x y + = 6 b i B1 B1 for clear explanation One solution, as one equation is not a multiple of the other and they are not parallel. ii B1 B1 for clear explanation None gradient is the same but the –the intercepts are different so they are parallel. iii B1 B1 for clear explanation Infinite number of solutions as the first equation is a multiple of the second, so drawing graphs gives the same line. c i M2 M1 for arranging equations in a suitable format y 2x = – –5 M1 equating both equations y = 0.5x + 1 Substitute for y in the first equation: A1 A1 cao 0.5x + 1 2x = – –5 = –1.5x –6 x = 4 Substitute into second equation: y = 0.5 × 4 + 1 = 3 Check by substituting both values in the first equation ii 3 2 × 4 = 3 8 = Correct. – – –5 iii No solution. d B1 B1 clear explanation Infinite number of solutions. 11 You can see how many times the graphs cross each other.
32 a b i ii iii c i ii iii d If one of the equations is a multiple of the other then there will be an infinite number of solutions, e.g. x + y = 5 2x + 2y = 10 Every point on the line x + y = 5 is a solution, giving an infinite number of solutions. If the equations have graphs that are parallel to each other then there will be no intersection and so no solution, e.g. x + y = 5 x + y = 6 One solution, as one equation is not a multiple of the other and they are not parallel. None –the gradient is the same but the intercepts are different so they are parallel. Infinite number of solutions as the first equation is a multiple of the second, so drawing graphs gives the same line. y – 2x = –5 y = 0.5x + 1 Substitute for y in the first equation: 0.5x + 1 – 2x = –5 –1.5x = –6 x = 4 Substitute into second equation: y = 0.5 × 4 + 1 = 3 Check by substituting both values in the first equation 3 – 2 × 4 = 3 – 8 = –5 Correct. No solution. Infinite number of solutions. You can see how many times the graphs cross each other. B1 B1 B1 B1 B1 B1 B1 M2 A1 B1 2 B1 for clear explanation B1 for use of a good example accompanying the explanation B1 for clear explanation B1 for use of a good example accompanying the explanation B1 for clear explanation B1 for clear explanation B1 for clear explanation M1 for arranging equations in a suitable format M1 equating both equations A1 cao B1 clear explanation M 11
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