SOLUTION: Solve the system of equations by using the elimination method. (If the system is dependent, enter a general solution in terms of c. If there is no solution, enter NO SOLUTION.) {3

Algebra ->  Inequalities -> SOLUTION: Solve the system of equations by using the elimination method. (If the system is dependent, enter a general solution in terms of c. If there is no solution, enter NO SOLUTION.) {3      Log On


   

Question 1130420: Solve the system of equations by using the elimination method. (If the system is dependent, enter a general solution in terms of c. If there is no solution, enter NO SOLUTION.)
{3x+ 4y= −8
x− 5y= −9
(x, y) =
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

3x%2B+4y=+-8
x-5y=+-9

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


Lets start with the given system of linear equations

3%2Ax%2B4%2Ay=-8
1%2Ax-5%2Ay=-9

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

Since the LCM of 3 and 1 is 3, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -3 like this:

1%2A%283%2Ax%2B4%2Ay%29=%28-8%29%2A1 Multiply the top equation (both sides) by 1
-3%2A%281%2Ax-5%2Ay%29=%28-9%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
3%2Ax%2B4%2Ay=-8
-3%2Ax%2B15%2Ay=27

Notice how 3 and -3 add to zero (ie 3%2B-3=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%283%2Ax-3%2Ax%29%2B%284%2Ay%2B15%2Ay%29=-8%2B27

%283-3%29%2Ax%2B%284%2B15%29y=-8%2B27

cross%283%2B-3%29%2Ax%2B%284%2B15%29%2Ay=-8%2B27 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:

19%2Ay=19

y=19%2F19 Divide both sides by 19 to solve for y



y=1 Reduce


Now plug this answer into the top equation 3%2Ax%2B4%2Ay=-8 to solve for x

3%2Ax%2B4%281%29=-8 Plug in y=1


3%2Ax%2B4=-8 Multiply



3%2Ax=-8-4 Subtract 4 from both sides

3%2Ax=-12 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%28-12%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=-4 Multiply the terms on the right side


So our answer is

x=-4, y=1

which also looks like

(-4, 1)

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

3%2Ax%2B4%2Ay=-8
1%2Ax-5%2Ay=-9

we get



graph of 3%2Ax%2B4%2Ay=-8 (red) 1%2Ax-5%2Ay=-9 (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 (-4,1). This verifies our answer.



answer:
(x,+y) =(-4,+1)