SOLUTION: 3r +5s =3 r + 2s =13

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Question 83041: 3r +5s =3
r + 2s =13
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Let r and s be x and y (they represent the same idea)
If you want to solve by using addition, then...

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


Lets start with the given system of linear equations




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:

Multiply the top equation (both sides) by 1
Multiply the bottom equation (both sides) by -3


So after multiplying we get this:



Notice how 3 and -3 add to zero (ie )


Now add the equations together. In order to add 2 equations, group like terms and combine them




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:



Divide both sides by to solve for y



Reduce


Now plug this answer into the top equation to solve for x

Plug in


Multiply



Subtract from both sides

Combine the terms on the right side

Multiply both sides by . This will cancel out on the left side.


Multiply the terms on the right side


So our answer is

,

which also looks like

(, )

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




we get



graph of (red) (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 (,). This verifies our answer.




If you want to solve by using substitution, then...
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations




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

Subtract from both sides

Divide both sides by 5.


Which breaks down and reduces to



Now we've fully isolated y

Since y equals we can substitute the expression into y of the 2nd equation. This will eliminate y so we can solve for x.


Replace y with . Since this eliminates y, we can now solve for x.

Distribute 2 to

Multiply



Reduce any fractions

Subtract from both sides


Make 13 into a fraction with a denominator of 5


Combine the terms on the right side



Make 1 into a fraction with a denominator of 5

Now combine the terms on the left side.


Multiply both sides by . This will cancel out and isolate x

So when we multiply and (and simplify) we get



<---------------------------------One answer

Now that we know that , lets substitute that in for x to solve for y

Plug in into the 2nd equation

Multiply

Add to both sides

Combine the terms on the right side

Multiply both sides by . This will cancel out 2 on the left side.

Multiply the terms on the right side


Reduce


So this is the other answer


<---------------------------------Other answer


So our solution is

and

which can also look like

(,)

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




we get


graph of (red) and (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (,). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (,) into the system of equations


Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies



Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies


Since the solution (,) satisfies the system of equations






this verifies our answer.
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