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# Module 6: Math 113

### Module 6: Solving Systems of Linear Equations by Addition and in Three Variables

1. Solve the system for x = {x+y=4 {x-y=8 First, eliminate the y variable by adding the equations together. Making the equation 2x=12. Devide both sides by 2 to eliminate 2x. Answer is: x = 12
Solve the system for y = {x+y=4 {12+y=4 Now you know x = 6 so plug it into one of the equations as the x-value. Solve for y. y = 4 - 12. making the answer y = -8. The solution for the system is (8,4)
2. Solve the system = {-2x+y=2 {-x-3y=4 Multiply both sides of the first equation by 3. Making the equation -3(-2x+y)= -3(2). Simplifies to 6x-3y=-6
Solve for; {6x-3y=-6 {-x-3x=y Add the resulting eqations and divide by 5, giving you the answer x = -2
Solve for Y = {-2 + y = 2 Plug in -2 for x. Making the equation -2(-2)+y=2. Giving you 4+y=2. Subtract 4 from both sides. Your answer is y=-2
When do you multiply both sides by a chosen number? EVERY time when solving by the addition method.
Solving for true statements. {3x-2y=2 {-9x+6y=-6 Multiply both sides of the first equation by 3, giving you {3(3x-2y)=3(2). Simplify and add both equations. Anser becomes 0=0, a TRUE statement.
Solving systems of two linear equations by the addition method. Solve this system; -x - y/2 =5/2 - x/2+y/4 = 0 First we multiply both equations by their lowest common denominator which, in this case, is 2 and 4. making them {-2x-y=5 {-2x+y=0
Now finish solving {-2x-y=5 {-2x+y=0 add the equations to get -4x=5.
Sove for Y from the previous equation. Solve by eliminating X. To do this, Multiply each side of the first equation by -1. to make it {-1(-2x-y)=-1(5). simplify then add the two equations. your answer is y= -5/2
Solve the System = {3x-y+z=-15 {x+2y-z=1 {2x+3y-2z=0 Add equations 1 and 2 to eliminate Z and create equation 4.Use 1 and 3 to eliminate Z again by multilpying both sides of equation one by 2.Add this to equation 3 to get equation 5. Now multiply both sides of equation 4 by -1 simplify and add to equation 5
Solve the System = {2x-4t+8z=2 {-x-3y+z=11 {x-2y+4z=0 Add 1 & 2 to eliminate Z and get eq 4. Add 1 & 3 to eliminate Z again and get eq 5. Add 4 & 5 to solve for X & Y.
Solve the system; {6x-3y+12z=4 {-6x+4y-2z=7 {-2x+y-4z=3 Add 1 & 2 to eliminate Z and get eq 4. Add 1 & 3 to eliminate Z again and get eq 5. Add 4 & 5 to solve for X & Y.
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