Systems of Equations

There is a way where you can tell if the systems of equations has one solution, infinite solutions, or no solutions.  When one solution is graphed, it has two lines that intersect at one common point. No solutions is when two lines are parallel and will never intersect. Infinite solutions is when it looks like only one line on the graph, when it is two of the same line that overlap.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

When you look at the systesms of equations in slope-interceot form, one solution has a different slope, no solutions have the same slope but different y-intercepts, and infinite solutions have the same slope and y-intercept. For the inspection method, you must know the other form of the linear equation, called standard form, which is Ax + By = C. When it is in standard form, you won't be able to clearly tell what the slope and y-intercept is.  The inspection method is to help you find the slope and y-intercept.  Inspection method is when you use the formula -A/B in each equation to see if this coefficient ratio is equivalent or non-equivalent. 

 

2x + 1y = 5

4x + 2y = 10

-2/1 = -4/2

Equivalent

 

If the ratios are equivalent, then they are either a system with no solutions or infinite solutions.  If they aren't equivalent, there is only one solution and you have to solve the systems from there.  To find out if the equivalent coefficient ratios are are infinite or no solutions, you have to determine the y-intercept.  Once you find the y-intercept, then you can tell whether it is infinite solutions or no solutions using what we learned previously.  As a reminder, infinite solutions have the same equations and no solutions have the same slope but different y-intercepts. 

 

Infinite solutions:

2(2x + y = 5) =

4x + 2y = 10