1. Chapter 4 Section 4.1 Solving Systems of Equations in Two Variables

2. Example: -3 x + 2 y = 4 4 x + y = 13 This has solutions of: x = 2 and y =5 Substitute: -3(2) + 2(5) = -6 + 10 = 4 4(2) + 5 = 8 + 5 =13 Write the solution to this system as: (2,5) We will discuss three methods to find the solution to a system of equations 1. Graphing 2. Substitution 3. Elimination

3. Solving by Graphing As the name would imply the graphs of linear equations are lines. The idea is to graph both lines on the same graph carefully. Look at the point where the two lines cross (or try to estimate it as best as you can) the x and y coordinates are the simultaneous solutions to the system of equations. Look at the previous example: slope is 3/2 y -intercept is 2 slope is -4 y -intercept is 13 The problem that you run into with graphing to find the solutions is that it can be very imprecise. When the solutions involve fractions for example reading exactly where the lines cross is hard to tell on the graph. This is why we will look at two other methods algebraic methods that tell you the simultaneous solutions. The coordinates of the point the lines cross are (2,5)

4. More graphing examples: x = -3 Vertical line at -3 y = 2 Horizontal line at 2 Solution (-3,2) More graphing examples: slope is 3 y -intercept is -4 slope is 3 y -intercept is -1 These lines are parallel which means they do not intersect. This means there is no simultaneous solution to the system of equations. A system of equations that has no simultaneous solution we call inconsistent or no solution.

5. Substitution Method This gives the simultaneous solution by changing from 2 equations with 2 variables to one equation with one variable and solving that. Then take the value you found and plug into the substituted equation to find the other value. This will give the exact values for the simultaneous solution. This is the first example we started with. Use substitution to solve it. 1. Pick one of the equations and solve it for one of the variables. (In this case we solved the 2 nd equation for y.) 2. Substitute for the variable you solved for into the other equation. (We substituted into the 1 st equation for y.) 3. Solve the equation by isolating the variable. This will give you the value for one of the variables. 4. Substitute back into the solved equation from step 1. 5. Put the two together to form the simultaneous solution.

6. Solve each of the systems of equations below by substitution: step 1 step 2 step 3 step 4 step 5 This is a good method to use when one of the equations is easy to solve for one of the variables step 1 step 2 step 3 This is a false statement. This means the equations has no solution. This means there is no solution to simultaneous system of equations. The system is inconsistent. (Note) If you get a true statement it means the lines are the same and the number of solutions is infinite.

7. Elimination Method The method again find the exact values of a simultaneous solution. The idea is to multiply one or both equations by number so that if you add them together one of the variables drops out (or is eliminated). Consider the following example. 1. Multiply second equation by -2. (This gives the y variable the same coefficient but opposite in sign.) 2. Add the equations so now the y drops out. 3. Solve for the remaining variable. 4. Plug back into one of the remaining equations. 5. Solve to find the other variable. 6. Simultaneous Solution.

8. 1. Multiply first equation by 3 the second by -5 so the coefficient of x is the same number but different signs. 2. Add the equations together so x drops out. 3. In this case y is solved for already. 4. Plug back into one of the original equations. 5. Solve for x. 6. Solution of the simultaneous system. Solve this simultaneous system

9. Larger Systems In general a system of equations that has more than two variables is difficult to understand. The graphs of these equations are hard to visualize. There are some things we know about them. We will get only one solution (unique) if the number of variables and the number of equations is the same. For example if there are 4 variables w, x, y and z and 4 equations this system can have a single solution. It might have more than one or not any though. If the number of variables is less than the number of equations there will never be just one solution (i.e. unique). There will either be an infinite number of different solutions or none at all. This system might have a unique solution. This system will not have a unique solution.