Systems of Linear Equations

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Mrs. Jarin PromsriMrs. Panchaporn KantayasakunMrs. Khrongsri NampoonSystem of Linear EquationsHow to: solve by graphing, substitution, linear combinations, and special types of linear systems What is a Linear System, Anyways?A linear system includes two, or more, equations, and each includes two or more variables.When two equations are used to model a problem, it is called a linear system.

Before You BeginImportant Terms to knowLinear system: two equations that form one equationSolution: the answer to a system of linear equation; must satisfy both equations***: a solution is written as an ordered pair: (x,y)Leading Coefficient: any given number that is before any given variable (for example, the leading coefficient in 3x is 3.)Isolate: to get alone

Ways to Solve Linear SystemsBy: SubstitutionSolving Linear Systems by SubstitutionBasic steps:1. Solve one equation for one of its variables2. Substitute that expression into the other equation and solve for the other variable3. Substitute that value into first equation; solve4. Check the solution Example: The Substitution MethodHeres the problem:Equation one-x+y=1Equation two2x+y=-2

First, solve equation one for y y = x+1Next, substitute the above expression in for y in equation two, and solve for xHeres how: Equation two 2x+y=-2 2x+ (x+1)=-2 3x+1=-2 3x=-3 x=-1Congratulations! You now know x has a value of 1but you still need to find y.To do so First, write down equation one y=x+1 y= (-1)+1 y=0So, now what? Youre done; simply write out the solution as (-1,0)***Did you remember?To write a solution, once youve found x and y, you must put x first and then y: (x,y)Solving Linear Systems by Linear Combinations11Solving Systems by means of Linear CombinationsBasic steps:1. Arrange the equations with like terms in columns2. After looking at the coefficients of x and y, you need to multiply one or both equations by a number that will give you new coefficients for x or y that are opposites.3. Add the equations and solve for the unknown variable4. Substitute the value gotten in step 3 into either of the original equations; solve for other variable5. Check the solution in both original equationsExample: Solving Systems by Linear CombinationsSolve this linear equation:Equation One: 3x+5y = 6Equation Two: -4x+2y =5Solve the linear systemEquation 1: 3x+5y=6Equation 2: -4x+2y=53x+5y= 6 ---------4x+2y= 5 --------4;4(3x+5y)= 46 12x+20y= 24 --------3 ;3(-4x+2y) = 35-12x +6y= 15 --------+ ; 12x+ 20y -12x + 6y = 24 + 15 26y = 39

Equation 2: -4x+2y=5Substitute the value you just found for

-4x+2( ) = 5 -4x+3 = 5 -4x = 2 x = The solution to the example system is ( )

A Final way to Solve Systems:Graph and CheckTypes of Solutions of Systems of EquationsOne solution the lines cross at one point

No solution the lines do not cross

Infinitely many solutions the lines coincide

An Example of the Quick graph on and Check MethodHeres the problem:Equation one-x+y=1Equation two2x+y=-2

Step 1 Download Application Quick graph from programe App Store.Step 2 open App Quick graph on Iphone or samsung etc.

Step 3 Click the plus sign.

Step 4 Type the equation in the form y=1+x and click Done .

Step 5 will have a graph

Step 6 Click the plus sign. And Type the equation in the form y=-2-2xand click Done

Step 7 will have a graph for equation y=-2-2x

Answer to the equation is the graph intersect (-1,0)Fun, Fun: Exercises

1. Solve the following Linear SystemEquation one: 3x-4y=10Equation two: 5x+7y=32. Solve the following Linear systemEquation one: x-6y=-19Equation two: 3x-2y=-93. Solve the following Linear systemEquation one: x+3y=7Equation two: 4x-7y=-104. Use linear combinations to solve this systemEquation one: x+2y=5Equation two: 3x-2y=75. Use linear combinations to solve this systemEquation one: 3x-5y=-4Equation two: -9x+7y=8Answers to the Exercises1. (2,-1)2. (-1,3)3. (1,2)4. (3,1)5. (-0.5, 0.5)

Check out the answers from of the Quick graph

3x-4y=105x+7y=31.Check out the answers from of the Quick graph2.

3x-2y=-9x-6y=-19Check out the answers from of the Quick graph3.

x+3y=74x-7y=-10Check out the answers from of the Quick graph4.

x+2y=53x-2y=7Check out the answers from of the Quick graph5.

-9x+7y=83x-5y=-4The End