| Solved by pluggable solver: Solve the System of Equations by Graphing |
Start with the given system of equations: In order to graph these equations, we need to solve for y for each equation. So let's solve for y on the first equation Now lets graph So let's solve for y on the second equation Now lets add the graph of From the graph, we can see that the two lines are identical (one lies perfectly on top of the other) and intersect at all points of both lines. So there are an infinite number of solutions and the system is dependent. |
| Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition |
Lets start with the given system of linear equations In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa). So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero. So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 2 and 5 to some equal number, we could try to get them to the LCM. Since the LCM of 2 and 5 is 10, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -2 like this: So after multiplying we get this: Notice how 10 and -10 add to zero (ie Now add the equations together. In order to add 2 equations, group like terms and combine them So after adding and canceling out the x terms we're left with: Now plug this answer into the top equation So our answer is which also looks like ( Notice if we graph the equations (if you need help with graphing, check out this solver) we get and we can see that the two equations intersect at ( |
| Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition |
Lets start with the given system of linear equations In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa). So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero. So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and -2 to some equal number, we could try to get them to the LCM. Since the LCM of 1 and -2 is -2, we need to multiply both sides of the top equation by -2 and multiply both sides of the bottom equation by -1 like this: So after multiplying we get this: Notice how -2 and 2 add to zero, -10 and 10 add to zero, -20 and 20 and to zero (ie So we're left with which means any x or y value will satisfy the system of equations. So there are an infinite number of solutions So this system is dependent |