Systems of Equations

OBJECTIVES To understand what a system of equations is. Be able to solve a system of equations from graphing the equations Determine whether the system has one solution, no solution, or an infinite amount of solutions. Be able to graph equations without using a graphing calculator.

Defining a System of Equations A grouping of 2 or more equations, containing one or more variables. x + y = 2 2x + y = 5 2y = x + 2 y = 5x - 7 6x - y = 5

How do we “solve” a system of equations??? By finding the point where two or more equations, intersect. x + y = 6 y = 2x 6 4 Point of intersection 2 1 2

How do we “solve” a system of equations??? By finding the point where two or more equations, intersect. x + y = 6 y = 2x 6 4 (2,4) 2 1 2

ax + by = c 2x + 3y = 6 ax + by = c -2x -2x 3y = 6 - 2x 3 y = 2 - 2 3 (Standard Form) -2x -2x WE WANT THIS FORM!!! 3y = 6 - 2x 3 y = 2 - 2 3 x y = - + 2 2 3 x y = mx + b (Slope- Intercept)

Non-Unique Solutions No Solution: when lines of a graph are parallel since they do not intersect, there is no solution the slopes are the same and the equation must be in slope-intercept form Slides 10 through 14 show how I explained the different solutions to their worksheet that the students worked on in class. I didn’t focus to much on the solution they found on the worksheet but rather on the type of solution or the concepts they the solutions involved. In each slide I explain each type of solution as well as how a system has these types of solutions. Not focusing to much on the powerpoint I referred back to their worksheet to one of the examples and asked students to remember how we knew the equations graphed were parallel by only looking at the equations. From their prior knowledge students knew that it was because the equations in the system contained the same slope; making connections with new and old material.

Example of No Solution 2x+y=8 y=-2x-3 Remember when graphing the equations need to be in slope intercept form!!!

Non-Unique Solutions Infinite Solutions: a pair of equations that have the same slope and y-intercept. also call a Dependent System Again giving explanations as to how we have such a solution. Just like the last slide, I also questioned students as to how we can find the whether a system has infinite solutions by looking at the equation, triggering prior knowledge.

Example of Infinite Solutions 2x+4y=8 4x+8y=16

Unique Solutions One Solution: the lines of two equations intersect also called an Independent System Though not a Non-Unique Solution, I explained to students that we do not call this solution as being non-unique. I included it in this slide as I wanted to students to understand the different between three types of solutions.

Graphing Systems in Slope-Intercept Form 2y + x = 8 y = 2x + 4 Steps: 1. Get both equations in slope-intercept form 2. Find the slope and y intercept of eq. 1 then graph 3. Find the slope and y intercept of eq. 2 then graph 4. Find point of intersection

Example 1: 2y+x=8 and y=2x+4 Equation 1 Equation 2 2y + x = 8 y = 2x + 4 -x -x (0,4) m= 2/1 B=4 2y= 8 -1x 2 2 2 The solution to the system is (0,4) Notice the slopes and y intercepts are different so there will only be one solution m= -1/2 b=4

Example 2: y=-6x+8 and y+6x=8 Equation 1 Equation 2 y=-6x+8 y+6x=8 b=8 -6x -6x m=-6(down) 1(right) y=8-6x b=8 Notice both equations have the same intercept and slope. This means all the points are solutions. m=-6(down) 1(right) Infinite Solution

Example 3: x-5y=10 and -5y=-x+40 Equation 1 Equation 2 x-5y=10 -5y=-x+40 __ __ ___ -x -x -5 -5 -5 -5y=10-1x __ __ __ -5 -5 -5 b=-8 m=1/5 b=-2 The slopes are the same with different intercepts so the lines are parallel m=1/5 No Solution

You Try Examples Determine whether the following equations have one, none, or infinite solutions by graphing on the graph paper provided. 1) 2) 2 3) x + 2y = 6 y = x - 1 3 x + 2y = 8 y = 3 ANS: One Solution (6,3) ANS: No Solution ANS: Infinite Solutions

Equation 1 Slope Intercept Form Equation 2 8x+3y=24 Infinite Solution

Equation1 Equation 2 Notice the slopes are the same and the y-intercepts are different. This means that the lines are parallel so they will never intersect No Solution

Equation 1 Equation 2 y=3 Notice the equations have different slopes and y-intercepts so the lines will have 1 solution Solution (6,3)

Algebraically determine if a point is a solution We must always verify a proposed solution algebraically. We propose (1,6) as a solution, so now we plug it in to both equations to see if it works: y = 6x and y = 2x + 4, 6 = 6(1) and 6=2(1)+4, (6)= 6 and 6= 6. Yes, (1,6) Satisfies both equations!

Algebraically determine if a point is a solution Determine if (2,12) is a solution to the system. y=6x and y=2x+4 12=6(2) and 12=2(2)+4 12=12 and 12=8 Yes No (2,12) is NOT a solution to the system