Question 86174
a)

*[invoke solving_linear_system_by_substitution 5, 2, 16, 3, -5, -9]



b)
{{{x^2-3y^2=13}}}
{{{x-2y=1}}}



{{{x^2-3y^2=13}}}
{{{x=1+2y}}} Solve the 2nd equation for x



{{{(1+2y)^2-3y^2=13}}} Plug in {{{x=1+2y}}}


{{{1+4y+4y^2-3y^2=13}}} Foil


{{{1+4y+4y^2-3y^2-13=0}}} Subtract 13 from both sides


{{{4y+y^2-12=0}}} Combine like terms


{{{y^2+4y-12=0}}} Rearrange the terms




Starting with the general quadratic


{{{ay^2+by+c}}}


the general form of the quadratic equation is:


{{{y = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}


So lets solve {{{y^2+4*y-12}}}


{{{y = (-4 +- sqrt( (4)^2-4*1*-12 ))/(2*1)}}} Plug in a=1, b=4, and c=-12




{{{y = (-4 +- sqrt( 16-4*1*-12 ))/(2*1)}}} Square 4 to get 16




{{{y = (-4 +- sqrt( 16+48 ))/(2*1)}}} Multiply {{{-4*-12*1}}} to get {{{48}}}




{{{y = (-4 +- sqrt( 64 ))/(2*1)}}} Combine like terms in the radicand (everything under the square root)




{{{y = (-4 +- 8)/(2*1)}}} Simplify the square root




{{{y = (-4 +- 8)/2}}} Multiply 2 and 1 to get 2


So now the eypression breaks down into two parts


{{{y = (-4 + 8)/2}}} or {{{y = (-4 - 8)/2}}}


Lets look at the first part:


{{{y=4/2}}} Add the terms in the numerator

{{{y=2}}} Divide


So one answer is

{{{y=2}}}

Now lets look at the second part:


{{{y=-12/2}}} 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(2)=1}}} Plug in {{{y=2}}}
{{{x=5}}} solve for x

So we have the solution (5,2)



{{{x-2(-6)=1}}} Plug in {{{y=-6}}}
{{{x=-11}}} solve for x

So we have the solution (-11,-6)


Heres some visual proof



{{{drawing( 500, 500, -15, 10, -10, 15, 
graph( 500, 500, -15, 10, -10, 15, sqrt((-13+x^2)/3),-sqrt((-13+x^2)/3), (x-1)/2),
circle(5,2,0.05),
circle(5,2,0.08), 
circle(-11,-6,0.05),
circle(-11,-6,0.08))
)}}} Graphs of {{{x^2-3*y^2=13}}} (hyperbola) and {{{x-2*y=1}}}(line) with the intersections (5,2) and (-11,-6)



Check:
Plug in (5,2)
{{{5^2-3*2^2=13}}}
{{{5-2*2=1}}}


{{{25-12=13}}}
{{{5-4=1}}}


{{{13=13}}}
{{{1=1}}} solution works


Plug in (-11,-6)
{{{(-11)^2-3*(-6)^2=13}}}
{{{(-11)-2*(-6)=1}}}


{{{121-108=13}}}
{{{-11+12=1}}}


{{{13=13}}}
{{{1=1}}} solution works