Question 288664
{{{4x^2 + y^2 - 64 = 0}}} Start with the second equation.



{{{y^2 = -4x^2+64}}} Solve for {{{y^2}}} by getting every other term to the right side.



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{{{4x^2 - 56x + 9y^2 + 160 = 0}}} Move back to the first equation.



{{{4x^2 - 56x + 9(-4x^2+64) + 160 = 0}}} Replace each {{{y^2}}} term with {{{ -4x^2+64}}} (since the two are essentially the same or equivalent).



{{{4x^2 - 56x -36x^2+576 + 160 = 0}}} Distribute.



{{{-32x^2 - 56x+736 = 0}}} Combine like terms.



Now let's solve {{{-32x^2 - 56x+736 = 0}}} by use of the quadratic formula.



*[invoke quadratic_formula -32,-56,736, "x"]



Since the solutions in terms of 'x' are {{{x=-23/4}}} or {{{x=4}}}, we can use them to find the corresponding solutions in terms of 'y'.



So if {{{x=-23/4}}}, then...



{{{y^2 = -4x^2+64}}} Start with the given equation



{{{y^2 = -4(-23/4)^2+64}}} Plug in {{{x=-23/4}}}



{{{y^2 = -4(529/16)+64}}} Square {{{-23/4}}} to get {{{-529/16}}}



{{{y^2 = -2116/16+64}}} Multiply



{{{y^2 = -529/4+64}}} Reduce.



{{{y^2 = -273/4}}} Combine like terms.



{{{y = ""+-sqrt(-273/4)}}} Take the square root of both sides.



Since the square root of a negative number is not a real number, this means that there are no real solutions of 'y' when {{{x=-23/4}}}. So we can ignore this value.



Now if {{{x=4}}}, then...



{{{y^2 = -4x^2+64}}} Start with the given equation



{{{y^2 = -4(4)^2+64}}} Plug in {{{x=4}}}



{{{y^2 = -4(16)+64}}} Square 4 to get 16.



{{{y^2 = -64+64}}} Multiply



{{{y^2 = 0}}} Combine like terms.



{{{y=sqrt(0)}}} Take the square root of both sides.



{{{y=0}}} Take the square root of 0 to get 0.



So when {{{x=4}}}, {{{y=0}}} giving us the ordered pair (4,0)



This means that the two graphs intersect at the only point of (4,0)