All categories

3 years ago

Graph the line given by 5x+6y=12 and the circle given by x²+y²=1.Find all solutions to the system of equations.

Mathematics

1 answer:

kenny6666 [7]3 years ago

3 0

Answer:

The system has no real solution.

Step-by-step explanation:

Hi there!

We have the following system of equations:

x² + y² = 1

5x + 6y = 12

The solutions of the system are those (x,y) pairs that satisfy both equations.

Let´s take the second equation and solve it for y:

5x + 6y =12

subtract 5x to both sides of the equation

6y = 12 - 5x

divide by 6 both sides

y = (12- 5x)/6

Now, let´s replace the y in the first equation:

x² + y² = 1

x² + [(12- 5x)/6]² = 1

Apply distributive property

x² + (12 - 5x)²/36 = 1

x² + (12 - 5x)(12 - 5x)/36 = 1

Apply distributive property

x² + (144 - 120x + 25x²)/36 = 1

Apply distributive property

x² + 4 - 10/3 · x + 25/36 · x² = 1

subtract 1 to both sides

61/36 · x² - 10/3 · x + 3 = 0

Using the quadratic formula, let´s find the solutions to this quadratic equation:

a = 61/36

b = -10/3

c = 3

[-b ± √(b² - 4ac)]/2a

[10/3 ± √(100/9 - 4(61/36)(3))]/2(61/36)

(10/3 ± √-83/9)/61/18

This expression has no real solution because √-83/9 is not defined in the set of real numbers. This means that both curves don´t intersect (see attached figure).

Have a nice day!

You might be interested in

3.7 (negative 83 x minus 3) = 3.7 (7 x minus 30)

kirill [66]

Answer:

x= 3 over 10 =0.300

Step-by-step explanation:

Hope this helps!

3 0

3 years ago

_______are the possible results of an experiment

Virty [35]

Outcome

Hope this helps :)

6 0

3 years ago

Evaluate the following limit:

Makovka662 [10]

If we evaluate the function at infinity, we can immediately see that:

$\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle L = \lim_{x \to \infty}{\frac{(x^2 + 1)^2 - 3x^2 + 3}{x^3 - 5}} = \frac{\infty}{\infty}} \end{gathered}$}$

Therefore, we must perform an algebraic manipulation in order to get rid of the indeterminacy.

We can solve this limit in two ways.

By comparison of infinities:

We first expand the binomial squared, so we get

$\large\displaystyle\text{$\begin{gathered}\sf \displaystyle L = \lim_{x \to \infty}{\frac{x^4 - x^2 + 4}{x^3 - 5}} = \infty \end{gathered}$}$

Note that in the numerator we get x⁴ while in the denominator we get x³ as the highest degree terms. Therefore, the degree of the numerator is greater and the limit will be \infty. Recall that when the degree of the numerator is greater, then the limit is \infty if the terms of greater degree have the same sign.

Dividing numerator and denominator by the term of highest degree:

$\large\displaystyle\text{$\begin{gathered}\sf L = \lim_{x \to \infty}\frac{x^{4}-x^{2} +4 }{x^{3}-5 } \end{gathered}$}\\$

$\ \ = \lim_{x \to \infty\frac{\frac{x^{4} }{x^{4} }-\frac{x^{2} }{x^{4}}+\frac{4}{x^{4} } }{\frac{x^{3} }{x^{4}}-\frac{5}{x^{4}} } }$

$\large\displaystyle\text{$\begin{gathered}\sf \bf{=\lim_{x \to \infty}\frac{1-\frac{1}{x^{2} } +\frac{4}{x^{4} } }{\frac{1}{x}-\frac{5}{x^{4} } } \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{1}{0}=\infty } \end{gathered}$}$

Note that, in general, 1/0 is an indeterminate form. However, we are computing a limit when x →∞, and both the numerator and denominator are positive as x grows, so we can conclude that the limit will be ∞.

5 0

2 years ago

Please answer ASAP!!!

drek231 [11]

Hi there!
------------------------
question - A 36-foot flagpole casts a 10 3/4 foot shadow, while the hospital nearby casts a 37 5/8 foot shadow. Find the height of the hospital
------------------------------------------------------------------------------------------------------------
if you correctly work this out the answer is:
The height of the hospital is 35 3/4 feet
--------------------------------------------------------------
Have a nice day!

6 0

3 years ago

When you add their numbers together you get 207 . Jen's number is 9 more than Carrie's, and Fran's number is 3 less than Jen's n

murzikaleks [220]

Answer: Fran's number is 70

Step-by-step explanation:

Let x represent Jen's number.

Let y represent Carrie's number.

Let z represent Fran's number.

When you add their numbers together you get 207. This means that

x + y + z = 207 - - - - - - - - - -1

Jen's number is 9 more than Carrie's number. It means that

y = x - 9

Fran's number is 3 less than Jen's number. It means that

z = x - 3

Substituting y = x - 9 and z = x - 3 into equation 1, it becomes.

x + x - 9 + x - 3 = 207

3x - 12 = 207

3x = 207 + 12 = 219

x = 219/3 = 73

y = x - 9 = 73 - 9

y = 64

z = x - 3 = 73 - 3

z = 70

8 0

3 years ago