Defining “System” of Equations• A grouping of 2 or more equations, containing one or more variables. x+y=2 2x + y = 5
How do we “ solve ” a system of equations?• By finding the point where two or more equations intersectx+y=6y = 2x 6 4 Point of intersection 2 (2,4) 1 2
There are 3 methods that we can use to Solve a “System” of EquationsToday we are going to focuson solving by graphing!
TODAY WE ARE GOING TOFOCUS ON “GRAPHING” TO SOLVE A SYSTEM OF EQUATIONS
System of Equations: “One Solution”One Solution: • the lines of two equations intersect The solution is the POINT that they intersect at.
1) Decide what form the equations are in: • What are the three? A. Slope-Intercept Form B. Standard Form C. Point-Slope Form2) Graph both lines.3) Determine the point of intersection and write this point as an ordered pair.
Graph the system of equations. Determine whetherthe system has one solution, no solution, orinfinitely many solutions. If the system has onesolution, determine the solution. Step 1- what form are the linear equations in? 2 y= x - 2 y = − x + 3 3 Slope-Intercept Form Use the slope and the y- intercept to graph.
y=x–2 2 y = − x + 3 3Step 2: Use the slope and y-intercept of each line to plot twopoints for each line on the same graph. y Line #1: Slope = 1 and the y-int = -2 Line #2: x Slope = -2/3 and the y-int. = 3 The point of intersection of the two lines is the point (3,1). This system of equations has one solution, the point (3,1)
y The two equations in slope-intercept form are: y = −x + 3 y = 2x − 6 x Plot points for each line. Draw in the lines.This system of equations represents two intersecting lines.The solution to this system of equations is a single point(3,0) .
System of Equations: “Solutions”“No Solutions” “Infinite Solutions”• when lines of a graph are • a pair of equations that parallel have the same slope and • since they do not intersect, y-intercept. there is no solution •What is happening here??
y The two equations in slope-intercept form are: 1 y = − x + 1 3 x 3 9 1 y = − x + or y = − x + 1 9 9 3 Plot points for each line. Draw in the lines.These two equations represent the same line.Therefore, this system of equations hasinfinitely many solutions .
With Your Buddy:On the graph paper I have given you: You and your elbowpartner graph and solve the system of linear equations.Determine whether the following equations have one, none, orinfinite solutions. If “one solution” graph it and give the pointof intersection. 1) y= x-1 2) y=3 ANS: No Solution, theANS: One Solution lines are parallel (6,3)
In-Class Practice 6-1Determine whether the following have one, none, or infinitesolutions by looking at the slope and y-intercepts, and grapheach system.
Closure• Write a 4 sentence summary, in your own words, on Systems of Equations. – Remember, a summary is how you would explain it to one of your friends if they asked you.