Lecture provides knowledge of the integrals yielding logarithmic and exponential functions. This chapter presents the following content: Integrals of f(x) = 1/x and of the other circular functions, integrals of exponential functions, the natural logarithmic function: A rigorous approach. | General Mathematics ADE 101 Unit 2 LECTURE No. 15 SIMULTANEOUS LINEAR EQUATIONS AND THEIR SOLUTION Today’s Objectives Knowledge Test Systems of Equations A set of equations is called a system of equations. The solutions must satisfy each equation in the system. If all equations in a system are linear, the system is a system of linear equations, or a linear system. Systems of Linear Equations: A solution to a system of equations is an ordered pair that satisfy all the equations in the system. A system of linear equations can have: 1. Exactly one solution 2. No solutions 3. Infinitely many solutions 6 There are four ways to solve systems of linear equations: 1. By graphing 2. By substitution 3. By addition (also called elimination) 4. By multiplication Systems of Linear Equations: 7 When solving a system by graphing: Find ordered pairs that satisfy each of the equations. Plot the ordered pairs and sketch the graphs of both equations on the same axis. The coordinates of the point or points of intersection of the graphs are the solution or solutions to the system of equations. Solving Systems by Graphing: 8 Solving Systems by Graphing: Consistent Dependent Inconsistent One solution Lines intersect No solution Lines are parallel Infinite number of solutions Coincide-Same line Three possible solutions to a linear system in two variables: One solution: coordinates of a point No solutions: inconsistent case Infinitely many solutions: dependent case Linear System in Two Variables 10 2x – y = 2 x + y = -2 2x – y = 2 -y = -2x + 2 y = 2x – 2 x + y = -2 y = -x - 2 Different slope, different intercept! 11 3x + 2y = 3 3x + 2y = -4 3x + 2y = 3 2y = -3x + 3 y = -3/2 x + 3/2 3x + 2y = -4 2y = -3x -4 y = -3/2 x - 2 Same slope, different intercept!! x – y = -3 2x – 2y = -6 x – y = -3 -y = -x – 3 y = x + 3 2x – 2y = -6 -2y = -2x – 6 y = x + 3 Same slope, same intercept! Same equation!! There is a somewhat shortened way to determine what type (one solution, no solutions, infinitely many | General Mathematics ADE 101 Unit 2 LECTURE No. 15 SIMULTANEOUS LINEAR EQUATIONS AND THEIR SOLUTION Today’s Objectives Knowledge Test Systems of Equations A set of equations is called a system of equations. The solutions must satisfy each equation in the system. If all equations in a system are linear, the system is a system of linear equations, or a linear system. Systems of Linear Equations: A solution to a system of equations is an ordered pair that satisfy all the equations in the system. A system of linear equations can have: 1. Exactly one solution 2. No solutions 3. Infinitely many solutions 6 There are four ways to solve systems of linear equations: 1. By graphing 2. By substitution 3. By addition (also called elimination) 4. By multiplication Systems of Linear Equations: 7 When solving a system by graphing: Find ordered pairs that satisfy each of the equations. Plot the ordered pairs and sketch the graphs of both equations on the same axis. The coordinates of the point or .
