Question 129570: I got stuck on this system problem. I always ended up with something odd like "-5" by itself. Help!
-3x + 3y = 4
-x + y = 3
Found 2 solutions by jim_thompson5910, bucky:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Let's solve this system by using elimination
Start with the given system of equations:
Now in order to solve this system by using elimination/addition, we need to solve (or isolate) one variable. I'm going to solve for y.
In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for  , we would have to eliminate  (or vice versa).
So lets eliminate . In order to do that, we need to have both coefficients that are equal in magnitude but have opposite signs (for instance 2 and -2 are equal in magnitude but have opposite signs). This way they will add to zero. By adding to zero, they can be eliminated.
So to make the coefficients equal in magnitude but opposite in sign, we need to multiply both coefficients by some number to get them to an common number. So if we wanted to get  and  to some equal number, we could try to get them to the LCM.
Since the LCM of and is , we need to multiply both sides of the top equation by  and multiply both sides of the bottom equation by like this:
 Multiply the top equation (both sides) by
 Multiply the bottom equation (both sides) by
Distribute and multiply
Now add the equations together. In order to add 2 equations, group like terms and combine them
Combine like terms and simplify
 Notice how the x terms cancel out
 Simplify
Since this equation is never true regardless of what x or y is, there are no solutions. So this system is inconsistent.
Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website! There's a good reason why you were having difficulty. Let's see if we can find out why. You are
given the two equations:
.
-3x + 3y = 4 and
- x + y = 3
.
Suppose we decide to use variable elimination to solve these two equations. Let's multiply the
bottom equation (both sides and all terms) by 3. When we do that the set of equations becomes:
.
-3x + 3y = 4 and
-3x + 3y = 9
.
If you subtract these equations, everything on the left side disappears and the right side
subtraction results in -5. I'll bet that's exactly what you did. You can tell right away
why this happened, but let's take a different look at it. Let's convert the two equations to the
slope-intercept form of y = mx + b in which m (the multiplier of x) is the slope and b (the
constant) is the value on the y-axis where the graph intercepts the y-axis.
.
Let's work on the converting the first equation as follows:
.
-3x + 3y = 4 <=== add 3x to both sides. On the left side this cancels the -3x & you have:
3y = 3x + 4 <=== divide both sides (all terms) by 3 to get:
y = x + 4/3 <=== slope intercept form of the first equation.
.
Next let's work on converting the second equation as follows:
.
-3x + 3y = 9 <=== add 3x to both sides. On the left side this cancels the -3x & you have:
3y = 3x + 9 <=== divide both sides (all terms) by 3 to get:
y = x + 3 <=== slope intercept form of the second equation.
.
Look at both the slope intercept forms. In both of these equations the multiplier of the
x is 1. This means that the graphs of each of these equations has a slope of 1. And from
the constants you can tell that one crosses the x axis at +4/3 and the other crosses at
x = +3. Since the slopes are equal the graphs are parallel and are separated vertically.
To have a common solution the graphs must cross, but parallel lines do not cross. This
means that these two equations do not have a common solution, and that is why you were
having trouble figuring out what you were doing wrong. You weren't doing anything wrong ...
you just needed to understand what was going on. And just for grins, here's the graphs of the two
equations so you can see what I mean. The "rust red" graph is for the top equation, and
the green graph is for the bottom equation:
.
.
Hope this helps you to understand the problem a little better.
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