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Lucan Community College Leaving Certificate Mathematics Higher Level Mr Duffy Simultaneous Equations and Intersections © Ciarán Duffy Simultaneous Equations and Intersections Suppose we want to find where 2 lines meet. e.g. 1 y x 3 and y 2 x 5 Sketching the lines gives The point of intersection has an x-value between -1 and 0 and a y-value between 3 and 4. The exact values can be found by solving the equations simultaneously Simultaneous Equations and Intersections At the point of intersection, we notice that the x-values on both lines are the same and the yvalues are also the same. yy x 3 yy 2 x 5 23 , 113 As the y-values are the same, the right-hand sides of the equations must also be the same. x 3 2x 5 2 3x 23 x Substituting into one of the original equations, we can y x 3 find y: y 23 3 y 11 3 The point of intersection is 23 , 113 Simultaneous Equations and Intersections Sometimes the equations first need to be rearranged: e.g. 2 y 2 x 4 (1) 3 x y 11 (2) Solution: Equation (2) can be written as y 11 3 x (2a ) Now, eliminating y between (1) and (2a) gives: 2 x 4 11 3 x 5 x 15 x3 Substituting into (1): y 2 x 4 y2 The point of intersection is (3, 2) Simultaneous Equations and Intersections Exercises 1. Solution: Eliminate y: Find the point of intersection of the following pairs of lines: y 4 2 x (1) y x 5 (2) 4 2x x 5 9 3x y 2 Point of intersection is (3,2) 2 x y 7 (1) 2. y 3 x (2) y 7 2 x (1a ) Rearrange Solution: (1): Eliminate y: 3 x 7 2 x x 4 y 1 Point of intersection is (4,1) x3 Simultaneous Equations and Intersections 2 y x e.g. 3 Find the points of intersection of and y 3 2x y x2 There are 2 points of intersection y 3 2x We again solve the equations simultaneously but this time there will be 2 pairs of x- and y-values Simultaneous Equations and Intersections e.g. 1 y x2 (1) y 3 2 x (2) Since the y-values are equal we can eliminate y by equating the right hand sides of the equations: x 2 3 2 x This is a quadratic equation, so we get zero on one side and try 2 x 2x 3 0 to factorise: ( x 1)( x 3) 0 x 1 or x 3 To find the y-values, we use the linear equation, which in this example is equation (2) x 1 y 3 2(1) y 1 x 3 y 3 2(3) y 9 The points of intersection are (1, 1) and (-3, 9) Simultaneous Equations and Intersections Sometimes we need to rearrange the linear equation before eliminating y y x2 3 e.g. 2 (1) y 3 x 1 (2) Rearranging (2) gives y 3 x 1 ( 2a ) x 2 3 3x 1 x 2 3x 4 0 Eliminating y: y 3 x 1 (4, 13) ( x 1)( x 4) 0 x 1 or x4 ( 1, 2) Substituting in (2a): x 1 y 2 x 4 y 13 y x2 3 Simultaneous Equations and Intersections Exercise Find the points of intersections of the following curve and line y x 2 2 (1) x y8 ( 2) The solution is on the next slide Simultaneous Equations and Intersections y x 2 2 (1) x y8 ( 2) y 8 x ( 2a ) Rearrange (2): Solution: x2 2 8 x x2 x 6 0 ( x 3)( x 2) 0 x 3 Eliminate y: or Substitute in (2a): y 8 x x 3 y 8 ( 3) y 11 x2 y 8 ( 2) y6 The points of intersection are (3, 11) and x2 (2, 6) Simultaneous Equations and Intersections Special Cases e.g. 1 Consider the following equations: y x 2 2 (1) y x 1 (2) y x2 2 y x 1 The line and the curve don’t meet. Solving the equations simultaneously will not give any real solutions Simultaneous Equations and Intersections Suppose we try to solve the equations: y x 2 2 (1) y x 1 (2) Eliminate y: x2 2 x 1 x2 x 1 0 Calculating the discriminant, b 2 4ac we get: b 2 4ac (1) 2 4(1)(1) 14 3 0 b 4ac 0 The quadratic equation has no real roots. 2 Simultaneous Equations and Intersections y x 2 3 (1) y 4 x 1 (2) e.g. 2 Eliminate y: x 2 3 4 x 1 x2 4x 4 0 The discriminant, b 2 4ac 4 2 4(1)( 4) 0 The quadratic equation has equal roots. 2 Solving x 4 x 4 0 y x2 3 y 4 x 1 ( x 2)( x 2) 0 x 2 (twice) x 2 y 7 The line is a tangent to the curve. Simultaneous Equations and Intersections SUMMARY A linear and a quadratic equation represent a line and a curve. To solve a linear and a quadratic equation simultaneously: • Eliminate one unknown to give a quadratic equation in the 2nd unknown, e.g. ax 2 bx c 0 • • b 2 4ac 0 2 points of intersection b 2 4ac 0 the line is a tangent to the curve Substitute into the linear equation to find the values of the 1st unknown. Solve for the 2nd unknown b 2 4ac 0 the line and curve do not meet and the equations have no real solutions. Simultaneous Equations and Intersections Exercises Decide whether the following pairs of lines and curves meet. If they do, find the point(s) of intersection. For each pair, sketch the curve and line. 1. y x 2 3 y 2 x 2 2. y x 2 3 y 7 x 7 3. y x 1 0 y x2 3 Simultaneous Equations and Intersections Solutions 1. y x 2 3 y 2 x 2 x 2 3 2 x 2 x2 2x 1 0 b 2 4ac 4 4(1)(1) 0 b 2 4ac 0 the line is a tangent to the curve x2 2x 1 0 ( x 1)( x 1) 0 x 1 y4 y x2 3 y 2x 2 Simultaneous Equations and Intersections Solutions 2. y x 2 3 y 7 x 7 x2 3 7x 7 x 2 7 x 10 0 b 2 4ac 49 4(1)(10) 9 b 2 4ac 0 there are 2 points of intersection x 2 7 x 10 0 ( x 2)( x 5) 0 x 2, 5 x2 y7 x 5 y 28 y x2 3 y 7x 7 Simultaneous Equations and Intersections Solutions 3. y x 1 0 y x2 3 x2 3 x 1 x2 x 4 0 b 2 4ac ( 1) 2 4(1)(4) 15 b 2 4ac 0 there are NO points of intersection y x2 3 y x 1 0 Simultaneous Equations and Intersections Simultaneous Equations and Intersections The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet. Simultaneous Equations and Intersections Two Lines At the point of intersection, we notice that the x-values on both lines are the same and the y-values are the same. y x 3 y 2x 5 y 2x 5 y x 3 As the y-values are the same, the righthand sides of the equations must also be the same. x 3 2x 5 2 3x 23 x Substituting into one of the original equations, we can find y: y x 3 y 23 3 y 11 3 The point of intersection is 23 , 113 Simultaneous Equations and Intersections 1 quadratic equation and 1 linear equation e.g. y x2 (1) y 3 2 x (2) Since the y-values are equal we can eliminate y by equating the right hand sides of the equations: x 2 3 2 x This is a quadratic equation, x 2x 3 0 ( x 1)( x 3) 0 so we get zero on one side and try to factorise: 2 x 1 or x 3 To find the y-values, we use the linear equation, which in this example is equation (2) x 1 y 3 2(1) y 1 x 3 y 3 2(3) y 9 The points of intersection are (1, 1) and (-3, 9) Simultaneous Equations and Intersections Sometimes we need to rearrange the linear equation before eliminating y y x2 3 e.g. (1) y 3 x 1 (2) Rearranging (2) gives y 3 x 1 ( 2a ) x 2 3 3x 1 x 2 3x 4 0 Eliminating y: ( x 1)( x 4) 0 x 1 or x4 Substituting in (2a): x 1 y 2 x 4 y 13 Simultaneous Equations and Intersections Special Cases e.g. 1 Consider the following equations: y x 2 2 (1) y x 1 (2) The line and the curve don’t meet. Solving the equations simultaneously will not give any real solutions. The discriminant b 2 4ac 0 Simultaneous Equations and Intersections y x 2 3 (1) y 4 x 1 (2) e.g. 2 Eliminate y: x 2 3 4 x 1 x2 4x 4 0 The discriminant, b 2 4ac 4 2 4(1)( 4) 0 The quadratic equation has equal roots. 2 Solving x 4 x 4 0 ( x 2)( x 2) 0 x 2 (twice) x 2 y 7 The line is a tangent to the curve. Simultaneous Equations and Intersections SUMMARY A linear and a quadratic equation represent a line and a curve. To solve a linear and a quadratic equation simultaneously: • Eliminate one unknown to give a quadratic equation in the 2nd unknown, e.g. ax 2 bx c 0 • • b 2 4ac 0 2 points of intersection b 2 4ac 0 the line is a tangent to the curve Substitute into the linear equation to find the values of the 1st unknown. Solve for the 2nd unknown b 2 4ac 0 the line and curve do not meet and the equations have no real solutions.