Question 129570
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:
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-3x + 3y = 4 and
- x + y = 3
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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:
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-3x + 3y = 4 and
-3x + 3y = 9
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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.
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Let's work on the converting the first equation as follows:
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-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.
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Next let's work on converting the second equation as follows:
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-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.
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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:
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{{{graph(400,400,-10,10,-10,10,(1/3)*(3x + 4),x+3)}}}
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Hope this helps you to understand the problem a little better.
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