SOLUTION: y=1/3x + 2/3 y=5/7x-2

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Question 110680: y=1/3x + 2/3
y=5/7x-2
Found 2 solutions by stanbon, MathLover1:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
y=1/3x + 2/3
y=5/7x - 2
------------
(1/3)x+(2/3) = (5/7)x-2
[(5/7)-(1/3)]x = (2/3)+2
(8/21)x = 8/3
x = (8/3)(21/8)
x = 7
-------------
Substitute into y = (1/3)x+(2/3) to solve for "y":
y = (1/3)*7+2/3 = 9/3 = 3
===================
Cheers,
Stan H.

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
y=1%2F3x+%2B+2%2F3
y=5%2F7x%962
In standard form:
-%281%2F3%29x+%2B+y=+2%2F3

-%285%2F7%29x+%2B+y=+-2
Solution:
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition
TEST

%28%28-1%2F3%29%29%2Ax%2B%281%29%2Ay=%282%2F3%29 Start with the first equation


3%28%28%28-1%2F3%29%29%2Ax%2B%281%29%2Ay%29=%283%29%2A%28%282%2F3%29%29 Multiply both sides by the LCD 3



-1%2Ax%2B3%2Ay=2Distribute and simplify


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%28%28-5%2F7%29%29%2Ax%2B%281%29%2Ay=%28-2%29 Start with the second equation


7%28%28%28-5%2F7%29%29%2Ax%2B%281%29%2Ay%29=%287%29%2A%28%28-2%29%29 Multiply both sides by the LCD 7



-5%2Ax%2B7%2Ay=0 Distribute and simplify



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Lets start with the given system of linear equations

-1%2Ax%2B3%2Ay=2
-5%2Ax%2B7%2Ay=0

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 -5 to some equal number, we could try to get them to the LCM.

Since the LCM of -1 and -5 is 5, we need to multiply both sides of the top equation by -5 and multiply both sides of the bottom equation by 1 like this:

-5%2A%28-1%2Ax%2B3%2Ay%29=%282%29%2A-5 Multiply the top equation (both sides) by -5
1%2A%28-5%2Ax%2B7%2Ay%29=%280%29%2A1 Multiply the bottom equation (both sides) by 1


So after multiplying we get this:
5%2Ax-15%2Ay=-10
-5%2Ax%2B7%2Ay=0

Notice how 5 and -5 add to zero (ie 5%2B-5=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%285%2Ax-5%2Ax%29-15%2Ay%2B7%2Ay%29=-10%2B0

%285-5%29%2Ax-15%2B7%29y=-10%2B0

cross%285%2B-5%29%2Ax%2B%28-15%2B7%29%2Ay=-10%2B0 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

-8%2Ay=-10

y=-10%2F-8 Divide both sides by -8 to solve for y



y=5%2F4 Reduce


Now plug this answer into the top equation -1%2Ax%2B3%2Ay=2 to solve for x

-1%2Ax%2B3%285%2F4%29=2 Plug in y=5%2F4


-1%2Ax%2B15%2F4=2 Multiply



-1%2Ax%2B15%2F4=2 Reduce



-1%2Ax=2-15%2F4 Subtract 15%2F4 from both sides

-1%2Ax=8%2F4-15%2F4 Make 2 into a fraction with a denominator of 4

-1%2Ax=-7%2F4 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ax=%28-7%2F4%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.


x=7%2F4 Multiply the terms on the right side


So our answer is

x=7%2F4, y=5%2F4

which also looks like

(7%2F4, 5%2F4)

Notice if we graph the equations (if you need help with graphing, check out this solver)

-1%2Ax%2B3%2Ay=2
-5%2Ax%2B7%2Ay=0

we get



graph of -1%2Ax%2B3%2Ay=2 (red) -5%2Ax%2B7%2Ay=0 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (7%2F4,5%2F4). This verifies our answer.