SOLUTION: Please solve the system by addition and substitution methods. 3x-y=1 3x-y=2 I can determine that by looking at the equations that there is no solution. But I can not figure ou

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons  -> Linear Equations Lesson -> SOLUTION: Please solve the system by addition and substitution methods. 3x-y=1 3x-y=2 I can determine that by looking at the equations that there is no solution. But I can not figure ou      Log On


   

Question 120532: Please solve the system by addition and substitution methods.
3x-y=1
3x-y=2
I can determine that by looking at the equations that there is no solution. But I can not figure out how to do the math to prove my theory.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Substitution:

Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

3%2Ax-1%2Ay=1
3%2Ax-1%2Ay=2

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

-1%2Ay=1-3%2AxSubtract 3%2Ax from both sides

y=%281-3%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=-1%2B3%2Ax Now we've fully isolated y

Since y equals -1%2B3%2Ax we can substitute the expression -1%2B3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-1%2Ahighlight%28%28-1%2B3%2Ax%29%29=2 Replace y with -1%2B3%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-1%2A%28-1%29-1%283%29x=2 Distribute -1 to -1%2B3%2Ax

3%2Ax%2B1-3%2Ax=2 Multiply



3%2Ax%2B1-3%2Ax=2 Reduce any fractions

3%2Ax-3%2Ax=2-1 Subtract 1 from both sides


3%2Ax-3%2Ax=1 Combine the terms on the right side



0%2Ax=1 Now combine the terms on the left side.
0%2F1=1%2F1 Since this expression is not true, we have an inconsistency.


So there are no solutions. The simple reason is the 2 equations represent 2 parallel lines that will never intersect. Since no intersections occur, no solutions exist.


graph of 3%2Ax-1%2Ay=1 (red) and 3%2Ax-1%2Ay=2 (green) (hint: you may have to solve for y to graph these)


and we can see that the two equations are parallel and will never intersect. So this system is inconsistent





Elimination:


Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

3%2Ax-1%2Ay=1
3%2Ax-1%2Ay=2

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 3 and 3 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and 3 is 3, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -1 like this:

1%2A%283%2Ax-1%2Ay%29=%281%29%2A1 Multiply the top equation (both sides) by 1
-1%2A%283%2Ax-1%2Ay%29=%282%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
3%2Ax-1%2Ay=1
-3%2Ax%2B1%2Ay=-2

Notice how 3 and -3 and 1 and 1 add to zero (ie 3%2B-3=0 -1%2B1=0)

However 1 and -2 add to -1 (ie 1%2B-2=-1);


So we're left with

0=-1


which means no value of x or y value will satisfy the system of equations. So there are no solutions


So this system is inconsistent