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

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Question 83041: 3r +5s =3
r + 2s =13
Answer by jim_thompson5910(35256) About Me  (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

3%2Ax%2B5%2Ay=3
1%2Ax%2B2%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 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%2B5%2Ay%29=%283%29%2A1 Multiply the top equation (both sides) by 1
-3%2A%281%2Ax%2B2%2Ay%29=%2813%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
3%2Ax%2B5%2Ay=3
-3%2Ax-6%2Ay=-39

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%285%2Ay-6%2Ay%29=3-39

%283-3%29%2Ax%2B%285-6%29y=3-39

cross%283%2B-3%29%2Ax%2B%285-6%29%2Ay=3-39 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:

-1%2Ay=-36

y=-36%2F-1 Divide both sides by -1 to solve for y



y=36 Reduce


Now plug this answer into the top equation 3%2Ax%2B5%2Ay=3 to solve for x

3%2Ax%2B5%2836%29=3 Plug in y=36


3%2Ax%2B180=3 Multiply



3%2Ax=3-180 Subtract 180 from both sides

3%2Ax=-177 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%28-177%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=-59 Multiply the terms on the right side


So our answer is

x=-59, y=36

which also looks like

(-59, 36)

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

3%2Ax%2B5%2Ay=3
1%2Ax%2B2%2Ay=13

we get



graph of 3%2Ax%2B5%2Ay=3 (red) 1%2Ax%2B2%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 (-59,36). 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

3%2Ax%2B5%2Ay=3
1%2Ax%2B2%2Ay=13

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

5%2Ay=3-3%2AxSubtract 3%2Ax from both sides

y=%283-3%2Ax%29%2F5 Divide both sides by 5.


Which breaks down and reduces to



y=3%2F5-%283%2F5%29%2Ax Now we've fully isolated y

Since y equals 3%2F5-%283%2F5%29%2Ax we can substitute the expression 3%2F5-%283%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B2%2Ahighlight%28%283%2F5-%283%2F5%29%2Ax%29%29=13 Replace y with 3%2F5-%283%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B2%2A%283%2F5%29%2B2%28-3%2F5%29x=13 Distribute 2 to 3%2F5-%283%2F5%29%2Ax

1%2Ax%2B6%2F5-%286%2F5%29%2Ax=13 Multiply



1%2Ax%2B6%2F5-%286%2F5%29%2Ax=13 Reduce any fractions

1%2Ax-%286%2F5%29%2Ax=13-6%2F5 Subtract 6%2F5 from both sides


1%2Ax-%286%2F5%29%2Ax=65%2F5-6%2F5 Make 13 into a fraction with a denominator of 5


1%2Ax-%286%2F5%29%2Ax=59%2F5 Combine the terms on the right side



%285%2F5%29%2Ax-%286%2F5%29x=59%2F5 Make 1 into a fraction with a denominator of 5

%28-1%2F5%29%2Ax=59%2F5 Now combine the terms on the left side.


cross%28%285%2F-1%29%28-1%2F5%29%29x=%2859%2F5%29%285%2F-1%29 Multiply both sides by 5%2F-1. This will cancel out -1%2F5 and isolate x

So when we multiply 59%2F5 and 5%2F-1 (and simplify) we get



x=-59 <---------------------------------One answer

Now that we know that x=-59, lets substitute that in for x to solve for y

1%28-59%29%2B2%2Ay=13 Plug in x=-59 into the 2nd equation

-59%2B2%2Ay=13 Multiply

2%2Ay=13%2B59Add 59 to both sides

2%2Ay=72 Combine the terms on the right side

cross%28%281%2F2%29%282%29%29%2Ay=%2872%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.

y=72%2F2 Multiply the terms on the right side


y=36 Reduce


So this is the other answer


y=36<---------------------------------Other answer


So our solution is

x=-59 and y=36

which can also look like

(-59,36)

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

3%2Ax%2B5%2Ay=3
1%2Ax%2B2%2Ay=13

we get


graph of 3%2Ax%2B5%2Ay=3 (red) and 1%2Ax%2B2%2Ay=13 (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 (-59,36). This verifies our answer.


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Check:

Plug in (-59,36) into the system of equations


Let x=-59 and y=36. Now plug those values into the equation 3%2Ax%2B5%2Ay=3

3%2A%28-59%29%2B5%2A%2836%29=3 Plug in x=-59 and y=36


-177%2B180=3 Multiply


3=3 Add


3=3 Reduce. Since this equation is true the solution works.


So the solution (-59,36) satisfies 3%2Ax%2B5%2Ay=3



Let x=-59 and y=36. Now plug those values into the equation 1%2Ax%2B2%2Ay=13

1%2A%28-59%29%2B2%2A%2836%29=13 Plug in x=-59 and y=36


-59%2B72=13 Multiply


13=13 Add


13=13 Reduce. Since this equation is true the solution works.


So the solution (-59,36) satisfies 1%2Ax%2B2%2Ay=13


Since the solution (-59,36) satisfies the system of equations


3%2Ax%2B5%2Ay=3
1%2Ax%2B2%2Ay=13


this verifies our answer.