Math Problems

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ALG2 Solve System of Equations by Graphing and Substitution or Elimination

How to Solve a System of Two Linear Equations

Solve systems of equations

Solve systems of equations

When solving a system of equations we can take multiple approaches. You approach may be dependent on your teachers directions or simply by the easiest path to find your solution based on your problem. Either way it is important to understand that we can solve a system of equations using any of the methods taught in this post event though some methods may be easier for some problems than for others.

Lets start with what the solution represents for a system then work our way backwards into how we can obtain the solution. The solution of a system is the values that make both of your equations true. In a system of two linear equations we will have two variables that we will need to determine. Graphically the values that make an equation true can be represented as a coordinate point on a line. If we have two values that are true for two equations, then the points must be on both lines. Therefore the point represent the intersection of the two graphs.

Here are the steps you will want to follow when solving a system by graphing

  1. Rewrite each equation in slope intercept form and identify the slope and y-intercept
  2. Graph each equation separately using the y-intercept and slope
  3. Identify the x and y values of the coordinate point at the intersection
  4. Identify the solution based on the intersection. No intersection, no solution inconsistent system. Same line, infinite many solutions, consistent system, dependent.  One intersection point, one solution, consistent system, independent.

To further examine this point lets take a look at the videos I created to show you how to solve a system of equations by graphing.

Solve system of equations by graphing with one solution

Solve system of equations by graphing with infinite many or no solutions

While graphing is a great way to visualize the solution of a system of equations it is not always the best path to take. What if the solution point is not at two integers, or if it is a really difficult problem to graph. In those cases we look to graphing a system of equations Algebraically.

We will first solve systems of equations using the substitution method. Substitution is very helpful when at least one variable in your equation has a coefficient of 1 or -1. The steps for substitution are as follows

  1. Identify the variable you want to solve for that has a coefficient of 1 or -1.(If none exists choose the variable that will be easiest to isolate)
  2. Solve for your variable using inverse operations
  3. Plug in the quantity that your variable is equal to in replacement of the same variable in the other equation.
  4. Simplify your equation and solve for the variable(If done correctly you should be solving for the other variable at this time)
  5. Plug in the value of your variable into one of the equations to solve for your other variable(I like to plug the value into the equation I originally solved for to minimize my steps)
  6. Check your answer

Solve system of equations by substitution with one solution


Solve system of equations by substitution with infinite many or no solutions

If you do not have to solve by substitution and you are left with a system that does not have a variable with a coefficient of 1 or -1 then there is another process we can follow. The process is call elimination and the steps go like this:

  1. Reorder each equation so that the same variables align vertically with each other.
  2. Identify the variables that have the same coefficients(positive or negative) If none of the variables have the same coefficients then determine the coefficients that have the smallest LCM. To obtain the LCM for each coefficient you will need to multiply one or both equations by a multiplier(Remember to multiply every term in the equation by the multiplier)
  3. If the two coefficients of one of the variables are the same value but one positive and one negative, then add the two equations vertically.  If the two coefficients of one of the variables has the same value and sign then you will need to subtract the two equations vertically.
  4. Simplify and solve for the remaining variable
  5. Replace value of your variable you solved for into one of the original equations for that same variable.  Then solve for the other variable
  6. Check your answer

Solve system of equations by elimination with one solution

Solve system of equations by elimination with infinite many or no solutions