SOLUTION: Can you please help with these problems. I need to solve for all values of x and y. I tried but my solution just didn't work. a) 5x + 2y= 16 b) x^2-3y^2= 13 3

Algebra ->  Equations -> SOLUTION: Can you please help with these problems. I need to solve for all values of x and y. I tried but my solution just didn't work. a) 5x + 2y= 16 b) x^2-3y^2= 13 3      Log On


   

Question 86174: Can you please help with these problems. I need to solve for all values of x and y. I tried but my solution just didn't work.
a) 5x + 2y= 16 b) x^2-3y^2= 13
3x - 5y= -9 x-2y=1
Thank you so much for your help.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a)
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

5%2Ax%2B2%2Ay=16
3%2Ax-5%2Ay=-9

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

2%2Ay=16-5%2AxSubtract 5%2Ax from both sides

y=%2816-5%2Ax%29%2F2 Divide both sides by 2.


Which breaks down and reduces to



y=8-%285%2F2%29%2Ax Now we've fully isolated y

Since y equals 8-%285%2F2%29%2Ax we can substitute the expression 8-%285%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-5%2Ahighlight%28%288-%285%2F2%29%2Ax%29%29=-9 Replace y with 8-%285%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-5%2A%288%29-5%28-5%2F2%29x=-9 Distribute -5 to 8-%285%2F2%29%2Ax

3%2Ax-40%2B%2825%2F2%29%2Ax=-9 Multiply



3%2Ax-40%2B%2825%2F2%29%2Ax=-9 Reduce any fractions

3%2Ax%2B%2825%2F2%29%2Ax=-9%2B40Add 40 to both sides


3%2Ax%2B%2825%2F2%29%2Ax=31 Combine the terms on the right side



%286%2F2%29%2Ax%2B%2825%2F2%29x=31 Make 3 into a fraction with a denominator of 2

%2831%2F2%29%2Ax=31 Now combine the terms on the left side.


cross%28%282%2F31%29%2831%2F2%29%29x=%2831%2F1%29%282%2F31%29 Multiply both sides by 2%2F31. This will cancel out 31%2F2 and isolate x

So when we multiply 31%2F1 and 2%2F31 (and simplify) we get



x=2 <---------------------------------One answer

Now that we know that x=2, lets substitute that in for x to solve for y

3%282%29-5%2Ay=-9 Plug in x=2 into the 2nd equation

6-5%2Ay=-9 Multiply

-5%2Ay=-9-6Subtract 6 from both sides

-5%2Ay=-15 Combine the terms on the right side

cross%28%281%2F-5%29%28-5%29%29%2Ay=%28-15%2F1%29%281%2F-5%29 Multiply both sides by 1%2F-5. This will cancel out -5 on the left side.

y=-15%2F-5 Multiply the terms on the right side


y=3 Reduce


So this is the other answer


y=3<---------------------------------Other answer


So our solution is

x=2 and y=3

which can also look like

(2,3)

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

5%2Ax%2B2%2Ay=16
3%2Ax-5%2Ay=-9

we get


graph of 5%2Ax%2B2%2Ay=16 (red) and 3%2Ax-5%2Ay=-9 (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 (2,3). This verifies our answer.


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

Plug in (2,3) into the system of equations


Let x=2 and y=3. Now plug those values into the equation 5%2Ax%2B2%2Ay=16

5%2A%282%29%2B2%2A%283%29=16 Plug in x=2 and y=3


10%2B6=16 Multiply


16=16 Add


16=16 Reduce. Since this equation is true the solution works.


So the solution (2,3) satisfies 5%2Ax%2B2%2Ay=16



Let x=2 and y=3. Now plug those values into the equation 3%2Ax-5%2Ay=-9

3%2A%282%29-5%2A%283%29=-9 Plug in x=2 and y=3


6-15=-9 Multiply


-9=-9 Add


-9=-9 Reduce. Since this equation is true the solution works.


So the solution (2,3) satisfies 3%2Ax-5%2Ay=-9


Since the solution (2,3) satisfies the system of equations


5%2Ax%2B2%2Ay=16
3%2Ax-5%2Ay=-9


this verifies our answer.






b)
x%5E2-3y%5E2=13
x-2y=1


x%5E2-3y%5E2=13
x=1%2B2y Solve the 2nd equation for x


%281%2B2y%29%5E2-3y%5E2=13 Plug in x=1%2B2y

1%2B4y%2B4y%5E2-3y%5E2=13 Foil

1%2B4y%2B4y%5E2-3y%5E2-13=0 Subtract 13 from both sides

4y%2By%5E2-12=0 Combine like terms

y%5E2%2B4y-12=0 Rearrange the terms



Starting with the general quadratic

ay%5E2%2Bby%2Bc

the general form of the quadratic equation is:

y+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve y%5E2%2B4%2Ay-12

y+=+%28-4+%2B-+sqrt%28+%284%29%5E2-4%2A1%2A-12+%29%29%2F%282%2A1%29 Plug in a=1, b=4, and c=-12



y+=+%28-4+%2B-+sqrt%28+16-4%2A1%2A-12+%29%29%2F%282%2A1%29 Square 4 to get 16



y+=+%28-4+%2B-+sqrt%28+16%2B48+%29%29%2F%282%2A1%29 Multiply -4%2A-12%2A1 to get 48



y+=+%28-4+%2B-+sqrt%28+64+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



y+=+%28-4+%2B-+8%29%2F%282%2A1%29 Simplify the square root



y+=+%28-4+%2B-+8%29%2F2 Multiply 2 and 1 to get 2

So now the eypression breaks down into two parts

y+=+%28-4+%2B+8%29%2F2 or y+=+%28-4+-+8%29%2F2

Lets look at the first part:

y=4%2F2 Add the terms in the numerator
y=2 Divide

So one answer is
y=2
Now lets look at the second part:

y=-12%2F2 Subtract the terms in the numerator
y=-6 Divide

So another answer is
y=-6

So our solutions are:
y=2 or y=-6

Now solve for x:
x-2%282%29=1 Plug in y=2
x=5 solve for x
So we have the solution (5,2)


x-2%28-6%29=1 Plug in y=-6
x=-11 solve for x
So we have the solution (-11,-6)

Heres some visual proof


Graphs of x%5E2-3%2Ay%5E2=13 (hyperbola) and x-2%2Ay=1(line) with the intersections (5,2) and (-11,-6)


Check:
Plug in (5,2)
5%5E2-3%2A2%5E2=13
5-2%2A2=1

25-12=13
5-4=1

13=13
1=1 solution works

Plug in (-11,-6)
%28-11%29%5E2-3%2A%28-6%29%5E2=13
%28-11%29-2%2A%28-6%29=1

121-108=13
-11%2B12=1

13=13
1=1 solution works