SOLUTION: Solve each of the following systems by addition. If a unique solution does not exist, state whether the system is inconsistent or dependent. Problem 12 is reminiscent of Examples

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Question 90259: Solve each of the following systems by addition. If a unique solution does not exist, state whether the system is inconsistent or dependent. Problem 12 is reminiscent of Examples 1, 2 and 3 (pp 690-692) and Problem 20 is much like Example 4 (p 693)
12) 2x + 3y = 1
5x + 3y = 16


20) x + 5y = 10
-2x – 10y = -20

Answer by jim_thompson5910(35256) 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

2%2Ax%2B3%2Ay=1
5%2Ax%2B3%2Ay=16

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

Since the LCM of 2 and 5 is 10, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -2 like this:

5%2A%282%2Ax%2B3%2Ay%29=%281%29%2A5 Multiply the top equation (both sides) by 5
-2%2A%285%2Ax%2B3%2Ay%29=%2816%29%2A-2 Multiply the bottom equation (both sides) by -2


So after multiplying we get this:
10%2Ax%2B15%2Ay=5
-10%2Ax-6%2Ay=-32

Notice how 10 and -10 add to zero (ie 10%2B-10=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2810%2Ax-10%2Ax%29%2B%2815%2Ay-6%2Ay%29=5-32

%2810-10%29%2Ax%2B%2815-6%29y=5-32

cross%2810%2B-10%29%2Ax%2B%2815-6%29%2Ay=5-32 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:

9%2Ay=-27

y=-27%2F9 Divide both sides by 9 to solve for y



y=-3 Reduce


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

2%2Ax%2B3%28-3%29=1 Plug in y=-3


2%2Ax-9=1 Multiply



2%2Ax=1%2B9 Subtract -9 from both sides

2%2Ax=10 Combine the terms on the right side

cross%28%281%2F2%29%282%29%29%2Ax=%2810%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.


x=5 Multiply the terms on the right side


So our answer is

x=5, y=-3

which also looks like

(5, -3)

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

2%2Ax%2B3%2Ay=1
5%2Ax%2B3%2Ay=16

we get



graph of 2%2Ax%2B3%2Ay=1 (red) 5%2Ax%2B3%2Ay=16 (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 (5,-3). This verifies our answer.


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


Lets start with the given system of linear equations

1%2Ax%2B5%2Ay=10
-2%2Ax-10%2Ay=-20

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 -2 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and -2 is -2, we need to multiply both sides of the top equation by -2 and multiply both sides of the bottom equation by -1 like this:

-2%2A%281%2Ax%2B5%2Ay%29=%2810%29%2A-2 Multiply the top equation (both sides) by -2
-1%2A%28-2%2Ax-10%2Ay%29=%28-20%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
-2%2Ax-10%2Ay=-20
2%2Ax%2B10%2Ay=20

Notice how -2 and 2 add to zero, -10 and 10 add to zero, -20 and 20 and to zero (ie -2%2B2=0) -10%2B10=0, and -20%2B20=0)


So we're left with

0=0


which means any x or y value will satisfy the system of equations. So there are an infinite number of solutions


So this system is dependent