SOLUTION: what is -x-10y=15 -x+4y=13 in elimination

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Question 969176: what is -x-10y=15
-x+4y=13 in elimination
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

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


Lets start with the given system of linear equations

-1%2Ax-10%2Ay=15
-1%2Ax%2B4%2Ay=13

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 -1 to some equal number, we could try to get them to the LCM.

Since the LCM of -1 and -1 is 1, 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%28-1%2Ax-10%2Ay%29=%2815%29%2A-1 Multiply the top equation (both sides) by -1
1%2A%28-1%2Ax%2B4%2Ay%29=%2813%29%2A1 Multiply the bottom equation (both sides) by 1


So after multiplying we get this:
1%2Ax%2B10%2Ay=-15
-1%2Ax%2B4%2Ay=13

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


Now add the equations together. In order to add 2 equations, group like terms and combine them
%281%2Ax-1%2Ax%29%2B%2810%2Ay%2B4%2Ay%29=-15%2B13

%281-1%29%2Ax%2B%2810%2B4%29y=-15%2B13

cross%281%2B-1%29%2Ax%2B%2810%2B4%29%2Ay=-15%2B13 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:

14%2Ay=-2

y=-2%2F14 Divide both sides by 14 to solve for y



y=-1%2F7 Reduce


Now plug this answer into the top equation -1%2Ax-10%2Ay=15 to solve for x

-1%2Ax-10%28-1%2F7%29=15 Plug in y=-1%2F7


-1%2Ax%2B10%2F7=15 Multiply



-1%2Ax%2B10%2F7=15 Reduce



-1%2Ax=15-10%2F7 Subtract 10%2F7 from both sides

-1%2Ax=105%2F7-10%2F7 Make 15 into a fraction with a denominator of 7

-1%2Ax=95%2F7 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ax=%2895%2F7%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.


x=-95%2F7 Multiply the terms on the right side


So our answer is

x=-95%2F7, y=-1%2F7

which also looks like

(-95%2F7, -1%2F7)

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

-1%2Ax-10%2Ay=15
-1%2Ax%2B4%2Ay=13

we get



graph of -1%2Ax-10%2Ay=15 (red) -1%2Ax%2B4%2Ay=13 (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 (-95%2F7,-1%2F7). This verifies our answer.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

what is -x-10y=15
-x+4y=13 in elimination
- x - 10y = 15 ------- eq (i)

- x +  4y = 13 ------- eq (ii)

    - 14y =  2 ------- Subtracting eq (ii) from eq (i)

y+=+2%2F%28-+14%29, or highlight_green%28y+=+-+1%2F7%29

Substitute -+1%2F7 for y in any of the 2 original equations to determine the value of x