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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 )

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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] .

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Solve the system of nonlinear equations.

x 2 + y 2 = 10 x −3 y = −10

( −1 , 3 )

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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

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Saurabh Reply
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
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simplify each radical by removing as many factors as possible (a) √75
Jason Reply
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what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
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Need help with math
Peya
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litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
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Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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