Lecture 5 (solving simultaneous equations)
- 1. D. J. Jayamanne NSBM 1 Solving Simultaneous Equations
- 2. Simultaneous Equations After studying this section, you will be able to: • solve simultaneous linear equations by substitution • solve simultaneous linear equations by elimination • solve simultaneous linear equations using straight line graphs Substitution, Elimination and Use of Graphs are methods used in solving simultaneous equations. NSBM 2
- 3. Solving a system of linear equations 1. Method of Substitution Example: Solve the following system using the method of substitution. 2𝑥 + 𝑦 = 4 𝑥 − 𝑦 = −1 NSBM 3 Solve one of the equations for one unknown in terms of the other. Then, substitute that in the other equation. That will yield one equation in one unknown, which we can solve.
- 4. Method of Substitution Example: Solve the following system using the method of substitution. 2𝑥 + 𝑦 = 4 𝑥 − 𝑦 = −1 Express one variable in terms of the other in equation 1. 2𝑥 = 4 − 𝑦 𝑥 = 4−𝑦 2 Substitute that in equation 2. 𝑥 − 𝑦 = −1 4−𝑦 2 − 𝑦 = −1 4 − 𝑦 − 2𝑦 = −2 6 = 3𝑦 𝑦 = 2 NSBM 4
- 5. Method of Substitution Since the value of y is known (y=2), you could find the value of x by using one of the equations. 2𝑥 + 𝑦 = 4 When y=2, 2𝑥 + 2 = 4 𝑥 = 1 Answer: 𝑥 = 1 , 𝑦 = 2 NSBM 5
- 6. Method of Elimination Example: Solve the same system using the method of elimination. 2𝑥 + 𝑦 = 4 𝑥 − 𝑦 = −1 Eliminate one variable using the given two equations. For e.g. if the above two equations are added, variable y could be eliminated. 2𝑥 + 𝑦 = 4 𝑥 − 𝑦 = −1 3𝑥 = 3 𝑥 = 1 Substitute 𝑥 = 1 in one of the equations to find y. 2𝑥 + 𝑦 = 4 2 1 + 𝑦 =4 𝑦 = 2 NSBM 6 +
- 7. Graphical approach Example: Solve the same system using the graphical approach. 2𝑥 + 𝑦 = 4 𝑥 − 𝑦 = −1 Plot the above two linear equations (lines) on a graph paper -A line could be sketched by identifying the x, y intercepts. (Note: Only two points are required to plot a line) The solution to the system is identified by the intersection point of the above two lines (linear functions) Note: In order to plot a line at least points are needed. NSBM 7
- 8. NSBM 8
- 9. Solving a system of linear equations Solve simultaneously for x and y: 1. 𝑥 + 𝑦 = 10 𝑥 − 𝑦 = 2 2. 2𝑥 + 𝑦 = 10 𝑥 − 𝑦 = 2 3. 𝑥 + 2𝑦 = 8 2𝑥 − 𝑦 = 1 4. 3𝑥 + 4𝑦 = 24 4𝑥 + 3𝑦 = 25 NSBM 9
- 10. Q1 Solve these simultaneous equations 1. 4𝑥 + 𝑦 = 17 2𝑥 + 𝑦 = 9 2. 5𝑥 + 2𝑦 = 13 𝑥 + 2𝑦 = 9 3. 3𝑥 + 2𝑦 = 11 2𝑥 − 2𝑦 = 14 4. 3𝑥 − 4𝑦 = 17 𝑥 − 4𝑦 = 3 NSBM 10 5. 2𝑥 + 5𝑦 = 37 𝑦 = 11 − 2𝑥 6. 4𝑥 − 3𝑦 = 7 𝑥 = 13 − 3𝑦
- 11. Q2 Solve these simultaneous equations 1. 𝑥 + 𝑦 = 10 𝑥 − 𝑦 = 8 2. 4𝑥 + 3𝑦 = 7 −4𝑥 + 𝑦 = 5 3. 3𝑥 − 𝑦 = 8 𝑥 + 2𝑦 = 5 4. 2w − 3𝑧 = −1 3𝑤 + 4𝑧 = 24 NSBM 11 5. 4𝑥 − 5𝑦 = 8 −4𝑥 + 5𝑦 = −8 6. 2𝑥 = 5𝑦 + 4 2𝑥 − 5𝑦 = 6
- 12. Q3 Solve the following equations simultaneously 1. 𝑥 + 𝑦 = 3 𝑦 = 𝑥 + 5 2. 3𝑥 − 2𝑦 = 5 𝑥 = 4𝑦 − 5 3. 7𝑥 + 6𝑦 = −9 y = −2𝑥 + 1 4. 5𝑥 − 6𝑦 = −4 𝑥 = 𝑦 NSBM 12 5. 𝑥 + 3𝑦 = 4 𝑥 − 2𝑦 = −1 6. −2𝑥 − 𝑦 = −3 3𝑥 + 𝑦 = 0 7. 8𝑥 − 𝑦 = 15 3𝑥 + 4𝑦 = 10 8. 3𝑥 − 5𝑦 = 12 𝑥 + 2𝑦 = 4
- 13. Q4 Solve the following equations simultaneously 1. 3𝑥 + 5𝑦 = −3 𝑥 − 5𝑦 = −5 2. 2𝑥 − 4𝑦 = −4 𝑥 + 2𝑦 = 8 3. 7𝑥 − 6𝑦 = −1 𝑥 − 2𝑦 = −1 4. 2𝑥 − 𝑦 = 1 4𝑥 + 𝑦 = 8 NSBM 13 5. 6𝑥 + 3𝑦 = 1 𝑦 = −2𝑥 − 5 6. 4𝑥 − 4𝑦 = 8 𝑥 − 𝑦 = 2 7. 4𝑥 − 2𝑦 = 8 2𝑥 − 𝑦 = 4 8. 𝑦 = −3𝑥 + 2 6𝑥 + 2𝑦 = 1