SOLUTION: I don't understand this; 4x + 5y = -14 8x + 10y = -20

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Question 117571: I don't understand this;
4x + 5y = -14
8x + 10y = -20
Found 2 solutions by jim_thompson5910, bucky:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

4%2Ax%2B5%2Ay=-14
8%2Ax%2B10%2Ay=-20

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

5%2Ay=-14-4%2AxSubtract 4%2Ax from both sides

y=%28-14-4%2Ax%29%2F5 Divide both sides by 5.


Which breaks down and reduces to



y=-14%2F5-%284%2F5%29%2Ax Now we've fully isolated y

Since y equals -14%2F5-%284%2F5%29%2Ax we can substitute the expression -14%2F5-%284%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


8%2Ax%2B10%2Ahighlight%28%28-14%2F5-%284%2F5%29%2Ax%29%29=-20 Replace y with -14%2F5-%284%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

8%2Ax%2B10%2A%28-14%2F5%29%2B10%28-4%2F5%29x=-20 Distribute 10 to -14%2F5-%284%2F5%29%2Ax

8%2Ax-140%2F5-%2840%2F5%29%2Ax=-20 Multiply



8%2Ax-28-8%2Ax=-20 Reduce any fractions

8%2Ax-8%2Ax=-20%2B28Add 28 to both sides


8%2Ax-8%2Ax=8 Combine the terms on the right side



0%2Ax=8 Now combine the terms on the left side.
0%2F1=8%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 4%2Ax%2B5%2Ay=-14 (red) and 8%2Ax%2B10%2Ay=-20 (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

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
You have two unknowns and therefore you need at least two independent equations to be able to
solve for the two unknowns.
.
One of the ways to solve for two equations with two unknowns is to make one of the equations have
a term in it that is equal in size but has the opposite sign as the corresponding term in the
other equation. Then add the two equations together in vertical columns and one of the unknown
terms drops out. That leaves you with a new equation that has only one variable, and you can
solve for that variable. Once you get one of the variables, you can go back to one of the original
equations, substitute for that variable, and solve for the second variable. Sounds complicated,
but it's not too bad ... let's just do the problem step by step.
.
Given the two equations:
.
4x + 5y = -14
8x + 10y = -20
.
Let's try to get rid of the variable x. We can do that by multiplying the top equation
(both sides ... all terms) by -2. This changes the top equation to:
.
-8x - 10y = +28
.
and when we put this over the original bottom equation the pair of equations is then:
.
-8x - 10y = +28
8x + 10y = -20
.
Notice now what happens when we add these two equations vertically in columns. The -8x
in the top equation cancels the +8x in the bottom equation ... so there is no x term in
the resulting equation.
.
But this problem also has a trick in it. Notice that when you add the terms in the y column,
the -10y and the +10y also cancels out. So we can't solve this problem as we normally would.
.
Since both the x-terms and the y-terms cancel out and the two terms on the right side are different ...
these two equations are for parallel lines. On a graph they will never cross and therefore,
there is no point of the form (x,y) which will satisfy both equations.
.
Not a good problem to start with if you are having trouble understanding the concept of
solving two equations with two unknowns. Just remember the process above and post a new
example that might have a common solution.
.
Sorry for the confusion above, but you've seen an example of a set of two linear equations in which
both variables disappear if you try variable elimination. And if that ever happens to you
again you now know what it means.
.