5.1 Solving Systems of Linear Equations by Graphing To solve by graphing, graph both linear equations. This gives an approximate solution. Algebraic methods are more exact (next 2 sections). If the graphs intersect at one point the system is consistent and the equations are independent.

5.1 Solving Systems of Linear Equations by Graphing If the graphs are parallel lines, there is no solution and the solution set is . The system is inconsistent. If the graphs represent the same line, there are an infinite number of solutions. The equations are dependent.

5.2 Solving Systems of Linear Equations by Substitution Solving by substitution: Solve for a variable Substitute for that variable in the other equation Solve this equation for the remaining variable Put your solution back into either of the original equations to solve for the other variable Check your solution with the other equation

5.2 Solving Systems of Linear Equations by Substitution Example: From the first equation we get y=2x-7, so substituting into the second equation:

5.2 Solving Systems of Linear Equations by Substitution If when using substitution both variables drop out and you get something like: 10=6 The system inconsistent and there is no solution (parallel lines) If when using substitution both variables drop out and you get something like: 10=10 The system dependent and every solution of one line is also on the other (same lines)

5.3 Solving Systems of Linear Equations by the Addition Method Solving systems of equations by the addition method (a.k.a. elimination): Write equations in standard form (variables line up) Multiply one of the equations to get coefficients of one of the variables to be opposites Add (or subtract) equations – so that one variable drops out Solve for the remaining variable. Plug you solution back into one of the original equations and solve for the other variable.

5.3 Solving Systems of Linear Equations by the Addition Method Example: Multiply the second equation by 3 to get: Adding equations you get:

5.3 Solving Systems of Linear Equations by the Addition Method If when using elimination both variables drop out and you get something like: 10=6 The system is inconsistent and there is no solution (parallel lines) If when using elimination both variables drop out and you get something like: 10=10 The system is dependent and every solution of one line is also on the other (same lines)

5.4 Applications of Linear Systems of Equations Solving an applied problem by writing a system of equations: Determine what you are to find – assign variables Draw a diagram, figure or make a chart of information. Write the system of equations Solve the system using substitution or elimination Answer the question from the problem.

5.4 Applications of Linear Systems of Equations Mixture problem: How many ounces of a 5% solution must be added to a 20% solution to get 10 ounces of 12.5% solution. Let x = # ounces of 5% solution Let y = # ounces of 20% solution

5.4 Applications of Linear Systems of Equations Solution to mixture problem in 2 variables: