MAT150 Unit 4-4 -Systems of Equations Linear and Linear

Objectives  Solve systems of linear equations graphically  Solve systems of linear equations algebraically with the substitution method  Solve systems of linear equations algebraically by elimination  Model systems of equations to solve problems  Determine if a system of linear equations is inconsistent or dependent  Solve systems of nonlinear equations algebraically

System of Linear Equations A system of equations is a collection of one or more equations, each with one or more variables. A solution of a system of equations consists of values of the variables that are solutions to each equation of the system. To solve a system of equations means to find all solutions of the system.

Dependent and Inconsistent Systems Unique solution No solution Graphs are intersecting lines. Graphs are parallel lines; the system is inconsistent.

Dependent and Inconsistent Systems Many solutions Graphs are the same line; the system is dependent.

Solution by Substitution One equation is solved for a variable and that variable is replaced by the equivalent expression in the other equation.

Example

Solution by Elimination We rewrite one or both of the equations in an equivalent form that allows us to eliminate one of the variables by adding or subtracting the equations.

Example Use the elimination method to solve the system. Solution

Modeling Systems of Equations Solution of real problems sometimes requires us to create two or more equations whose simultaneous solution is the solution to the problem.

Example An investor has $200,000 to invest, part at 8% and the remainder at 3%. If her investment goal is to have an annual income of $13,600, how much should she put in each investment? Solution

Example (cont.) An investor has $200,000 to invest, part at 8% and the remainder at 3%. If her investment goal is to have an annual income of $13,600, how much should she put in each investment? Solution

Example Use the elimination method to solve each of the following systems, if possible. Verify the solution graphically. a.b. Solution a.

Example (cont) Use the elimination method to solve each of the following systems, if possible. Verify the solution graphically. a.b. Solution b..

Example Use substitution to solve the system. Solution

Example Solve the system Solution

Example (cont) Solve the system Solution Find the points of intersection. The solutions are (2, 0) and (–2, 8).