<< Chapter < Page Chapter >> Page >
In this section, you will:
  • Solve a system of nonlinear equations using substitution.
  • Solve a system of nonlinear equations using elimination.
  • Graph a nonlinear inequality.
  • Graph a system of nonlinear inequalities.

Halley’s Comet ( [link] ) orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse. Other comets follow similar paths in space. These orbital paths can be studied using systems of equations. These systems, however, are different from the ones we considered in the previous section because the equations are not linear.

Halley’s Comet (credit: "NASA Blueshift"/Flickr)

In this section, we will consider the intersection of a parabola and a line, a circle and a line, and a circle and an ellipse. The methods for solving systems of nonlinear equations are similar to those for linear equations.

Solving a system of nonlinear equations using substitution

A system of nonlinear equations    is a system of two or more equations in two or more variables containing at least one equation that is not linear. Recall that a linear equation can take the form A x + B y + C = 0. Any equation that cannot be written in this form in nonlinear. The substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes.

Intersection of a parabola and a line

There are three possible types of solutions for a system of nonlinear equations involving a parabola and a line.

Possible types of solutions for points of intersection of a parabola and a line

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

  • No solution. The line will never intersect the parabola.
  • One solution. The line is tangent to the parabola and intersects the parabola at exactly one point.
  • Two solutions. The line crosses on the inside of the parabola and intersects the parabola at two points.

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

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

Solving a system of nonlinear equations representing a parabola and a line

Solve the system of equations.

x y = −1 y = x 2 + 1

Solve the first equation for x and then substitute the resulting expression into the second equation.

x y = −1        x = y −1 Solve for  x .        y = x 2 + 1        y = ( y −1 ) 2 + 1 Substitute expression for  x .

Expand the equation and set it equal to zero.

y = ( y −1 ) 2    = ( y 2 −2 y + 1 ) + 1    = y 2 −2 y + 2 0 = y 2 −3 y + 2    = ( y −2 ) ( y −1 )

Solving for y gives y = 2 and y = 1. Next, substitute each value for y into the first equation to solve for x . Always substitute the value into the linear equation to check for extraneous solutions.

    x y = −1 x ( 2 ) = −1           x = 1 x ( 1 ) = −1           x = 0

The solutions are ( 1 , 2 ) and ( 0 , 1 ) , 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!

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana 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