<< Chapter < Page Chapter >> Page >

Could we have substituted values for y into the second equation to solve for x in [link] ?

Yes, but because x is squared in the second equation this could give us extraneous solutions for x .

For y = 1

y = x 2 + 1 y = x 2 + 1 x 2 = 0 x = ± 0 = 0

This gives us the same value as in the solution.

For y = 2

y = x 2 + 1 2 = x 2 + 1 x 2 = 1 x = ± 1 = ± 1

Notice that −1 is an extraneous solution.

Solve the given system of equations by substitution.

3 x y = −2 2 x 2 y = 0

( 1 2 , 1 2 ) and ( 2 , 8 )

Got questions? Get instant answers now!

Intersection of a circle and a line

Just as with a parabola and a line, there are three possible outcomes when solving a system of equations representing a circle and a line.

Possible types of solutions for the points of intersection of a circle and a line

[link] illustrates possible solution sets for a system of equations involving a circle and a line.

  • No solution. The line does not intersect the circle.
  • One solution. The line is tangent to the circle and intersects the circle at exactly one point.
  • Two solutions. The line crosses the circle and intersects it at two points.

Given a system of equations containing a line and a circle, find the solution.

  1. Solve the linear equation for one of the variables.
  2. Substitute the expression obtained in step one into the equation for the circle.
  3. Solve for the remaining variable.
  4. Check your solutions in both equations.

Finding the intersection of a circle and a line by substitution

Find the intersection of the given circle and the given line by substitution.

x 2 + y 2 = 5 y = 3 x −5

One of the equations has already been solved for y . We will substitute y = 3 x −5 into the equation for the circle.

x 2 + ( 3 x −5 ) 2 = 5 x 2 + 9 x 2 −30 x + 25 = 5 10 x 2 −30 x + 20 = 0

Now, we factor and solve for x .

10 ( x 2 3 x + 2 ) = 0 10 ( x 2 ) ( x 1 ) = 0 x = 2 x = 1

Substitute the two x -values into the original linear equation to solve for y .

y = 3 ( 2 ) −5 = 1 y = 3 ( 1 ) −5 = −2

The line intersects the circle at ( 2 , 1 ) and ( 1 , −2 ) , which can be verified by substituting these ( x , y ) values into both of the original equations. See [link] .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve the system of nonlinear equations.

x 2 + y 2 = 10 x −3 y = −10

( −1 , 3 )

Got questions? Get instant answers now!

Solving a system of nonlinear equations using elimination

We have seen that substitution is often the preferred method when a system of equations includes a linear equation and a nonlinear equation. However, when both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps. As an example, we will investigate the possible types of solutions when solving a system of equations representing a circle and an ellipse.

Possible types of solutions for the points of intersection of a circle and an ellipse

[link] illustrates possible solution sets for a system of equations involving a circle and an ellipse .

  • No solution. The circle and ellipse do not intersect. One shape is inside the other or the circle and the ellipse are a distance away from the other.
  • One solution. The circle and ellipse are tangent to each other, and intersect at exactly one point.
  • Two solutions. The circle and the ellipse intersect at two points.
  • Three solutions. The circle and the ellipse intersect at three points.
  • Four solutions. The circle and the ellipse intersect at four points.

Questions & Answers

1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
Practice Key Terms 4

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask