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

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) (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

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 2.

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 -5 to

Multiply

Reduce any fractions

Add to both sides

Combine the terms on the right side

Make 3 into a fraction with a denominator of 2

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

Subtract from both sides

Combine the terms on the right side

Multiply both sides by . This will cancel out -5 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.

Solve the 2nd equation for x

Plug in

Foil

Subtract 13 from both sides

Combine like terms

Rearrange the terms

Starting with the general quadratic

the general form of the quadratic equation is:

So lets solve

Plug in a=1, b=4, and c=-12

Square 4 to get 16

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root

Multiply 2 and 1 to get 2

So now the eypression breaks down into two parts

or

Lets look at the first part:

Add the terms in the numerator
Divide

So one answer is

Now lets look at the second part:

Subtract the terms in the numerator
Divide

So another answer is

So our solutions are:
or

Now solve for x:
Plug in
solve for x
So we have the solution (5,2)

Plug in
solve for x
So we have the solution (-11,-6)

Heres some visual proof

Graphs of (hyperbola) and (line) with the intersections (5,2) and (-11,-6)

Check:
Plug in (5,2)

solution works

Plug in (-11,-6)

solution works

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