Solving 2x2 Systems of Equations by Combination/Elimination

Description/Explanation/Highlights

Video Description

This video explains how to solve a 2×2 system of equations by the combination/elimination method.

Steps and Key Points to Remember

To solve a 2×2 system of equations by combination/elimination, follow these steps:

  1. This method is called combination/elimination because to solve, we will combine the two equations and in doing so, eliminate one of the variables allowing us to solve for the other.
  2. After solving for one variable, we will substitute the solution into one of the original equations to solve for the other variable.
  3. Usually the variable will not automatically be eliminated by combining so we will need to multiply one or both of the equations by a number to force the elimination of a variable.
  4. Let’s look at an example. We would like to use this method to solve the system: \(2x+5y=19\) and \(4x+3y=24\). Start by making sure the equations are in order with the variables on the left and the number on the right side of the equal sign and placing them one under the other.
\(2x+5y=19\)
\(\underline{4x+3y=24}\)
  1. Note that if we combined the two as they are, we would get 6x and 8y and neither would be eliminated, so we must look for a number or numbers that we could multiply one or both equations by that would eliminate a variable.
  2. If we multiplied the top equation by -2, then the coefficient in front of x would then be negative 4 and we could eliminate the x which is what we want!

\(-2(2x+5y=19)\longrightarrow -4x-10y=-38\)
\(\underline{\;\;\;\;\;\;4x+3y=24}\)

  1. Now write them under each other and combine.
  2. The x is eliminated and we can solve for y.
\( -4x-10y=-38\)
\(\underline{\;\;\;4x+\;\;3y=\;\;\;24}\)
\(\;\;\;\;\;\;\;\;\;\;\;-7y=-14\)
\(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;y=\;\;\;\;\;2\)
  1. Substitute the value you just found for y (2) into one of the original equations and solve for x. I chose the first one. \(2x+5(2)=19\rightarrow 2x+10=19\rightarrow 2x=9\rightarrow x=4.5\)
  2. The solution for the system is \((4.5, 2)\).
  3. You can check by plugging x and y into the original equations and making sure when you simplify that you get a true statement.
  4. Sometimes there’s not an obvious number that you can multiply one of the equations by that will eliminate one of the variables as in this example.
\(8x+3y=41\)
\(\underline{6x+5y=39}\)
  1. The solution is to pick one of the letters to eliminate (I picked y because the numbers were smaller) and multiply each equation by the coefficient of the variable in the opposite equation. In addition if the signs are the same, make one of the numbers you are multiplying by negative.
\(\;\;\;5(8x+3y=41)\longrightarrow\;\;\;40x+15y=\;\;\;205\)
\(\underline{-3(6x+5y=39)}\longrightarrow\underline{-18x-15y=-107}\)
  1. Now combine and eliminate as before to solve for x and then substitute x into one of the original equations to find y.
\(\;\;\;40x+15y=\;\;\;205\)
\(\underline{-18x-15y=-107}\)
\(\;\;\;22x\;\;\;\;\;\;\;\;\;\;=\;\;\;\;\;88\)
\(x=4\)
By substituting in the first equation:\(8(4)+3y=41\longrightarrow y=3\)

Here are some key points to keep in mind when solving a 2×2 system of equations by combination/elimination.

  • Put the equations in order and line up one under the other to attempt to combine and eliminate one of the variables.
  • Most systems will not eliminate one variable automatically and one or both equations may need to be multiplied by a number to cause one of the variables to be eliminated when combined.
  • By choosing carefully, you may be able to multiply only one of the equations by a number to eliminate a variable in the system.
  • If it is not possible to eliminate a variable by multiplying only one of the equations, you can always pick one of the variables and multiply each equation by the coefficient in front of the elimination variable in the opposite equation.
  • If the signs are the same, one of the numbers that are being multiplied by needs to be made negative.
  • After eliminating a variable, solve the remaining equation for the remaining variable.
  • Substitute the solution for the variable that you got back into one of the original equations and solve it to find the value of the other variable.
  • As with any method, the answer can always be checked by plugging the solution into both equations of the system to make sure that both get a true statement when simplified.

Video Highlights

  • 00:00 Introduction
  • 00:15 \(2x+5y=19\) and \(4x+3y=24\) system example solved by combination/elimination.
  • 03:05 \(8x+3y=41\) and \(6x+5y=39\) example of solving by combination/elimination
  • 06:31 Conclusion
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