8.1 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.

8.1 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.

8.2 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

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

8.2 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)

8.3 Solving Systems of Linear Equations by Elimination Solving systems of equations by 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.

8.3 Solving Systems of Linear Equations by Elimination Example: Multiply the second equation by 3 to get: Adding equations you get:

8.3 Solving Systems of Linear Equations by Elimination If when using elimination both variables drop out and you get something like: 10=6 The system 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 dependent and every solution of one line is also on the other (same lines)

8.4 Linear Systems of Equations in Three Variables Linear system of equation in 3 variables: Example:

8.4 Linear Systems of Equations in Three Variables Graphs of linear systems in 3 variables: Single point (3 planes intersect at a point) Line (3 planes intersect at a line) No solution (all 3 equations are parallel planes) Plane (all 3 equations are the same plane)

8.4 Linear Systems of Equations in Three Variables Solving linear systems in 3 variables: Eliminate a variable using any 2 equations Eliminate the same variable using 2 other equations Eliminate a different variable from the equations obtained from (1) and (2)

8.4 Linear Systems of Equations in Three Variables Solving linear systems in 3 variables: Use the solution from (3) to substitute into 2 of the equations. Eliminate one variable to find a second value. Use the values of the 2 variables to find the value of the third variable. Check the solution in all original equations.

8.5 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.

8.5 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

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