CHAPTER 3 LINEAR SYSTEMS3.1 Solving Systems Using Tables and Graphs

3.2 Solving Systems Algebraically

SYSTEM OF EQUATIONS

A system of equations is a set of two or more equations

A linear system consists of linear equations A solution of a system is a set of values for

the variables that makes all the equations true. (usually an ordered pair) Systems can be solved be various methods:

graphing, substitution, and elimination

SOLVING A SYSTEM BY GRAPHING

1. Write each equation in slope-intercept form2. Graph each line3. Find the point of intersection (this is your

solution)4. Check by substituting the values into both

equations

SOLVE EACH SYSTEM BY GRAPHING

CLASSIFYING SYSTEMS A system of two linear equations can be

classified by the number of solutions it has A consistent systems has at least one solution

An independent system has one solution A dependent system has infinitely many solutions

An inconsistent system has no solution

WITHOUT GRAPHING, CLASSIFY EACH SYSTEM AS INDEPENDENT, DEPENDENT, OR INCONSISTENT

1. Rewrite each equation into slope-intercept form

2. Compare the slopes and y-intercepts Different slopes: independent system Same slope and same y-intercept: dependent

system Same slope and different y-intercept:

inconsistent

WITHOUT GRAPHING, CLASSIFY EACH SYSTEM AS INDEPENDENT, DEPENDENT, OR INCONSISTENT

WITHOUT GRAPHING, CLASSIFY EACH SYSTEM AS INDEPENDENT, DEPENDENT, OR INCONSISTENT

SOLVING SYSTEMS BY SUBSTITUTION

1. Solve one equation for one of the variables

2. Substitute the expression into the other equation and solve

3. Substitute the solution into one of the original equations and solve for the remaining variable

4. Check the solution

SOLVING SYSTEMS BY SUBSTITUTION

Use when it is easy to isolate one of the variables

SOLVING BY ELIMINATION1. Rewrite both equations in

standard form2. Multiply one or both systems

by an appropriate non-zero number (note you want one variable to drop out in the next step)

3. Add the equations4. Solve for the variable5. Substitute the value into one

of the original equation and solve for the remaining variable

6. Check the solution

SOLVE BY ELIMINATION

SOLVING SYSTEMS WITHOUT UNIQUE SOLUTIONS

Solving a system algebraically can sometimes lead to infinitely many solutions and/or no solution If you get a true result: infinitely many solutions If you get a false result: no solution

EXAMPLE: SOLVE THE SYSTEM

EXAMPLE: SOLVE THE SYSTEM