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Chapter 4 simultaneous equations

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atiqah ayie
atiqah ayie

This document provides an overview of solving simultaneous equations between linear and quadratic equations with two unknowns. It includes 4 examples of solving simultaneous equations involving a linear equation equal to a quadratic equation. The examples show the steps of substituting one equation into the other and solving. The document also includes a chapter review with 6 practice problems involving simultaneous equations.

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Additional Mathematics Module Form 4 Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis Page | 41 CHAPTER 4- SIMULATANEOUS EQUATIONS 4.1 INTRODUCTION 1. In Form Three, we have learned about to solve simultaneous equations between two linear equations with two unknowns. 2. In Form Four, we will learn about to solve simultaneous equations between linear equation and non- linear equation which is quadratic equation with two unknowns. 4.2 SOLVING SIMULTANOUS EQUATIONS Example 1 : Given 6=− yx and yxx =+− 1072 . Solve the equation. Solution: 6=− yx 6−= xy yxx =+− 1072 Substitute into , 61072 −=+− xxx 01682 =+− xx 0)4)(4( =−− xx 4=x Substitute 4=x into , 64 −=y 2−=y Example 2: The straight line xy 26 −= intersects the curve 082 =−+ xyy at points A and B. Find the coordinates of points A and B. Solution: xy 26 −= 082 =−+ xyy 1 2 1 2 1 1 2
Additional Mathematics Module Form 4 Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis Page | 42 Substitute into , 08)26()26( 2 =−−+− xxx 082636244 22 =−−++− xxxx 028182 2 =+− xx 01492 =+− xx 0)2)(7( =−− xx 7=x or 2=x Substitute 7=x , 2=x into , (i) 7=x )7(26 −=y (7,-8) 8−=y (ii) 2=x )2(26 −=y (2, 2) 2=y The coordinates of point A and point B are (2, 2) and (7,-8) Example 3: Given A (-2h, 5k) is the solution to the simultaneous equations 522 =+ yx and 32 =+ yx , find the values of h and k. Solution: Method 1 From the coordinate (-2h, 5k),-2h is the value for x and 5k is the value for y. Hence change the terms x and y with -2h and 5k respectively. 5)5()2( 22 =+− kh 5254 22 =+ kh 3)5(2)2( =+− kh 3210 =− hk 2 310 − = k h Substitute into , 525) 2 310 (4 22 =+ − k k 1 2 1 2 1 2 1 simplify
Additional Mathematics Module Form 4 Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis Page | 43 525) 4 960100 (4 2 2 =+ +− k kk 525960100 22 =++− kkk 0460125 2 =+− kk 0)25)(225( =−− kk 0225 =−k or 025 =−k 25 2 =k or 5 2 =k Substitute 25 2 =k , 5 2 =k into , (i) 25 2 =k 2 3) 25 2 (10 − =h 10 1 1−=h (ii) 5 2 =k 2 3) 5 2 (10 − =h 2 1 =h Method 2 522 =+ yx 32 =+ yx yx 23 −= Substitute into , 5)23( 22 =+− yy 59124 22 =++− yyy 04125 2 =+− yy 0)2)(25( =−− yy 2 1 2 2 1 simplify
Additional Mathematics Module Form 4 Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis Page | 44 025 =−y or 02 =−y 5 2 =y or 2=y Substitute 5 2 =y , 2=y into , (i) 5 2 =y ) 5 2 (23 −=x 5 1 2=x (ii) 2=y )2(23 −=x 1−=x From the coordinate (-2h, 5k),-2h is the value for x and 5k is the value for y. Hence, (i) 5 1 22 =− h 12 −=− h 2 1 =h (ii) 5 2 5 =k 25 =k 25 2 =k 5 2 =k Example 4: Find the range of values of m so that the straight line 72 −= mxy intersects the curve 22 += xy at two different points. Solution: 72 −= mxy 22 += xy 2 10 1 1−=h Compare the actual value of x with the given 1 2
Additional Mathematics Module Form 4 Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis Page | 45 Substitute into 272 2 +=− xmx 0922 =+− mxx If and intersect at two points, 042 >− acb 0)9)(1(4)2( 2 >−− am 0364 2 >−m 092 >−m Let 092 =−m 92 =m 3±=m y -3 3 m If 092 >−m , the range of values of m is 3−<m , 3>m . CHAPTER REVIEW EXERCISE 1. Solve the following simultaneous equations: (a) 12=+ yx and 982 += xy (b) 113 =+ qp and 2)2)(1( =++ qp 2. Find the coordinates of the points of intersection of the straight line xy −=102 and the curve yxy 72 2 =+ . 3. The sum of two numbers is 7 and the sum of the square of two numbers is 29. Find the two numbers. 4. Given that (2, 1) is one of the solutions of the simultaneous equations 42 =+ pyx and 1122 =++ xkyx , where p and k are constants, find the value of p and of k and the other solution. 5. Solve the simultaneous equations 52 =− yx and 1162 +−= xxy . 6. Solve the equation 161232 2 =+=+ xyxyx . 1 2 1 2

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Chapter 4 simultaneous equations

  • 1. Additional Mathematics Module Form 4 Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis Page | 41 CHAPTER 4- SIMULATANEOUS EQUATIONS 4.1 INTRODUCTION 1. In Form Three, we have learned about to solve simultaneous equations between two linear equations with two unknowns. 2. In Form Four, we will learn about to solve simultaneous equations between linear equation and non- linear equation which is quadratic equation with two unknowns. 4.2 SOLVING SIMULTANOUS EQUATIONS Example 1 : Given 6=− yx and yxx =+− 1072 . Solve the equation. Solution: 6=− yx 6−= xy yxx =+− 1072 Substitute into , 61072 −=+− xxx 01682 =+− xx 0)4)(4( =−− xx 4=x Substitute 4=x into , 64 −=y 2−=y Example 2: The straight line xy 26 −= intersects the curve 082 =−+ xyy at points A and B. Find the coordinates of points A and B. Solution: xy 26 −= 082 =−+ xyy 1 2 1 2 1 1 2
  • 2. Additional Mathematics Module Form 4 Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis Page | 42 Substitute into , 08)26()26( 2 =−−+− xxx 082636244 22 =−−++− xxxx 028182 2 =+− xx 01492 =+− xx 0)2)(7( =−− xx 7=x or 2=x Substitute 7=x , 2=x into , (i) 7=x )7(26 −=y (7,-8) 8−=y (ii) 2=x )2(26 −=y (2, 2) 2=y The coordinates of point A and point B are (2, 2) and (7,-8) Example 3: Given A (-2h, 5k) is the solution to the simultaneous equations 522 =+ yx and 32 =+ yx , find the values of h and k. Solution: Method 1 From the coordinate (-2h, 5k),-2h is the value for x and 5k is the value for y. Hence change the terms x and y with -2h and 5k respectively. 5)5()2( 22 =+− kh 5254 22 =+ kh 3)5(2)2( =+− kh 3210 =− hk 2 310 − = k h Substitute into , 525) 2 310 (4 22 =+ − k k 1 2 1 2 1 2 1 simplify
  • 3. Additional Mathematics Module Form 4 Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis Page | 43 525) 4 960100 (4 2 2 =+ +− k kk 525960100 22 =++− kkk 0460125 2 =+− kk 0)25)(225( =−− kk 0225 =−k or 025 =−k 25 2 =k or 5 2 =k Substitute 25 2 =k , 5 2 =k into , (i) 25 2 =k 2 3) 25 2 (10 − =h 10 1 1−=h (ii) 5 2 =k 2 3) 5 2 (10 − =h 2 1 =h Method 2 522 =+ yx 32 =+ yx yx 23 −= Substitute into , 5)23( 22 =+− yy 59124 22 =++− yyy 04125 2 =+− yy 0)2)(25( =−− yy 2 1 2 2 1 simplify
  • 4. Additional Mathematics Module Form 4 Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis Page | 44 025 =−y or 02 =−y 5 2 =y or 2=y Substitute 5 2 =y , 2=y into , (i) 5 2 =y ) 5 2 (23 −=x 5 1 2=x (ii) 2=y )2(23 −=x 1−=x From the coordinate (-2h, 5k),-2h is the value for x and 5k is the value for y. Hence, (i) 5 1 22 =− h 12 −=− h 2 1 =h (ii) 5 2 5 =k 25 =k 25 2 =k 5 2 =k Example 4: Find the range of values of m so that the straight line 72 −= mxy intersects the curve 22 += xy at two different points. Solution: 72 −= mxy 22 += xy 2 10 1 1−=h Compare the actual value of x with the given 1 2
  • 5. Additional Mathematics Module Form 4 Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis Page | 45 Substitute into 272 2 +=− xmx 0922 =+− mxx If and intersect at two points, 042 >− acb 0)9)(1(4)2( 2 >−− am 0364 2 >−m 092 >−m Let 092 =−m 92 =m 3±=m y -3 3 m If 092 >−m , the range of values of m is 3−<m , 3>m . CHAPTER REVIEW EXERCISE 1. Solve the following simultaneous equations: (a) 12=+ yx and 982 += xy (b) 113 =+ qp and 2)2)(1( =++ qp 2. Find the coordinates of the points of intersection of the straight line xy −=102 and the curve yxy 72 2 =+ . 3. The sum of two numbers is 7 and the sum of the square of two numbers is 29. Find the two numbers. 4. Given that (2, 1) is one of the solutions of the simultaneous equations 42 =+ pyx and 1122 =++ xkyx , where p and k are constants, find the value of p and of k and the other solution. 5. Solve the simultaneous equations 52 =− yx and 1162 +−= xxy . 6. Solve the equation 161232 2 =+=+ xyxyx . 1 2 1 2