# 4.1 System of linear Equations Solving graphically Solving by substitution Solving by addition.

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4.1 System of linear Equations Solving graphically Solving by substitution Solving by addition

A system of linear equations Def: A system of linear equations is two more linear equations grouped together. Example: System of linear equations

A solution to a system of linear equations Def: A solution to a system of linear equations in two variables is an ordered pair (x, y) that satisfies both equations. Ex. Is (1, 6) a solution?

3 cases for two lines Case 1: Intersect in one point Case 2: Do not intersect Case 3 : Same line

CASE 1: Intersect in one point One solution (x, y)

CASE 2: Do not intersect Parallel lines No solution

CASE 3: Same line infinite solutions

Solving a system: graphically Graph both equations. Find the point of intersection if there is one. You may be able to see that the lines are parallel if the slopes match. Therefore there would be no solution.

Graphing Ex. #1 Put both in y = mx+b graph find the point of intersection Check

Graphing Ex. #2 Put both in y = mx+b graph find the point of intersection Check

Solving a system: Substitution 1. Isolate a variable in one of the equations. (Some times done) 2. Substitute one equation into the other. 3. Solve the equation from step 2. 4. Substitute and find the other variable. 5. Check your solution in both equations.

Substitution Ex. #1

Substitution Ex. #2

Solving a system: addition 1. Write both equations in standard from, that is ax + by = c 2. possibly us multiplication so when the two equations are added together one variable will cancel out. 3. Add the equations together. 4. Solve the result of step 3. 5. Substitute and find the other variable. 6. Check our solution in all equations