Systems of linear equations

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Introduction to system of linear equations A system of linear equations consists of 2 or more linear equations that use the same variables. Ex: 2x+3y=10 OR 5a+2b=14 4x-7y=7 a+6b=14

The solution to a system of equation is the point or points that make both or all of the equations true. Remember that a point is represented by an ordered pair (x,y)

Let’s look at an example- is the given ordered pair a solution to the given system? 3𝑥+7𝑦=12 7𝑥−𝑦=−4 (−3,3) 2𝑥−7=−𝑦 −5𝑥+13=𝑦 (2,3) No YES

There are 3 ways you can solve for a system of linear equations: Graphing Substitution Elimination

Infinitely many solutions When solving for a system of linear equations you can have 3 different types of solutions: One solution No solution Infinitely many solutions

To solve a system of linear equation by graphing I first need to Convert to slope intercept form State m and b for both equations Find point of intersection State the answer as an ordered pair

2𝑥−2𝑦=−8 2𝑥+2𝑦=4

2𝑥−2𝑦=−8 2𝑥+2𝑦=4 One solution: (-1,3)

𝑦=−2𝑥+5 𝑦=−2𝑥+1

𝑦=−2𝑥+5 𝑦=−2𝑥+1 No solution

𝑥+𝑦=−2 2𝑥−3𝑦=−9

𝑥+𝑦=−2 2𝑥−3𝑦=−9 One solution (-3,1)

2𝑥+𝑦=−2 2𝑥+2=−𝑦

2𝑥+𝑦=−2 2𝑥+2=−𝑦 Infinite solutions

Looking at the above graphs and solutions, I can tell that the system will have an infinite number of solutions when the lines have the ___________________________________________. The system will have no solutions when ____________________________________________. The system of equations will have one solution when_____________________________________________.

Looking at the above graphs and solutions, I can tell that the system will have an infinite number of solutions when the lines have the same y-intercept b, and the same slope m. The system will have no solutions when the lines have the same slope m, but different y-intercepts b. The systems will have one solution when the lines have different slopes m.

Solving Systems by substitution: Solve one equation for either variable (x= or y=) Substitute the expression from step one into the second equation Solve the second equation for the given variable Plug your solution back into the first equation Write your solution as an ordered pair Go back and check your solution in BOTH equations.

2x-3y=-24 X+6y=18 Solve one equation for either variable (x= or y=) Which equation is the easiest to solve for one variable?

2x-3y=-24 X+6y=18 Solve one equation for either variable (x= or y=) x+6y=18 -6y -6y X=18-6y

Substitute the expression from step one into the second equation 2x-3y=-24 X=18-6y 2 (18-6y)-3y=-24

Solve the second equation for the given variable 2 (18-6y)-3y=-24 36-12y-3y=-24 36-15y=-24 -15y=-60 Y=4

Plug your solution back into the first equation X+6y=18 Y=4 X+6(4)=18 X+24=18 X=-6

Write your solution as an ordered pair (-6,4)

Go back and check your solution in BOTH equations. 2x-3y=-24 X+6y=18 (-6,4) 2(-6)-3(4)=-24 -12-12=-24 -24=-24 -6+6(4)=18 -6+24=18 18=18 ✔️ ✔️