## 17Calculus Precalculus - Solving Nonlinear Systems of Equations

##### 17Calculus

Types of Systems of Equations

 Basics of linear systems with the same number of equations as unknowns - covered on the main linear systems page Linear with fewer equations than unknowns - covered on the dependent systems page Non-Linear - covered on this page

What Does It Mean to 'Solve'?

Like you did with linear systems, solving non-linear systems means to find the points that solve the both equations. In terms of graphs, these points are where the graphs intersect. We use the term non-linear when at least one of the equations is not a straight line.
Here is a great video where the instructor explains this in more detail.

### Michel vanBiezen - What Does It Mean to "Solve..."?

video by Michel vanBiezen

Getting Started

Solving systems of nonlinear equations can be done using techniques you have already learned with linear equations, substitution and elimination. Knowing which one to use is based on the form of the equations and, if you carefully look at the systems, usually one of the techniques will seem to work best. Here are some ideas to get you started.

1. If you have terms that with powers, try elimination first. It is best not to solve for a variable under a power. For example, if one of your equations is $$y=x^2$$, do not try to solve for x in order to use substitution. This will introduce a complexity that can lead to incomplete and incorrect answers.
2. If there is an obvious substitution, try to substitute cautiously. Sometimes substitution can get messy. Other times it can simplify the equations significantly.
3. You may end up with a quadratic. In this case, completing the square will help.

Difference From Linear Systems

Although we use the same techniques (substitution and elimination), we may end up with more than one solution. If you think about what is going on, this makes sense. For example, if we have a parabola and a line, intersection of the two curves may occur at two points. It is difficult to know just from the equations, how many points solve the system.

If you are allowed to, it helps to plot a graph on your calculator or computer and get a feel for what the graphs look like. This will help you know what to do when solving the equations.

Okay, now try your hand at some practice problems.

Practice

Unless otherwise instructed, find all the solutions to these nonlinear systems of equations, giving your answers in exact form.

These practice problems are divided into these sections.
1. Basic
2. Intermediate
3. Word Problems
4. Inequalities

Basic

$$x^2 + y^2 = 13$$ and $$2x + y = -1$$

Problem Statement

$$x^2 + y^2 = 13$$ and $$2x + y = -1$$

$$( 1.2, -3.4 )$$ and $$( -2, 3 )$$

Problem Statement

$$x^2 + y^2 = 13$$ and $$2x + y = -1$$

Solution

### Michel vanBiezen - 4116 video solution

video by Michel vanBiezen

$$( 1.2, -3.4 )$$ and $$( -2, 3 )$$

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$$x^2+4x-y=7$$ and $$2x-y=-1$$

Problem Statement

$$x^2+4x-y=7$$ and $$2x-y=-1$$

$$(-4, -7)$$ and $$(2, 5)$$

Problem Statement

$$x^2+4x-y=7$$ and $$2x-y=-1$$

Solution

### mattemath - 1781 video solution

video by mattemath

$$(-4, -7)$$ and $$(2, 5)$$

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$$-x+y=4 \text{ and } x^2+y=3$$

Problem Statement

$$-x+y=4 \text{ and } x^2+y=3$$

Solution

### mattemath - 1782 video solution

video by mattemath

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$$-2x+y=5 \text{ and } x^2+3x-y=1$$

Problem Statement

$$-2x+y=5 \text{ and } x^2+3x-y=1$$

Solution

### mattemath - 1783 video solution

video by mattemath

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$$x^2 - y^2 = 1$$ and $$x^2 + y^2 = 49$$

Problem Statement

$$x^2 - y^2 = 1$$ and $$x^2 + y^2 = 49$$

$$( 5, 2\sqrt{6} )$$, $$( 5, -2\sqrt{6} )$$, $$( -5, 2\sqrt{6} )$$ and $$( -5, -2\sqrt{6} )$$

Problem Statement

$$x^2 - y^2 = 1$$ and $$x^2 + y^2 = 49$$

Solution

### Michel vanBiezen - 4117 video solution

video by Michel vanBiezen

$$( 5, 2\sqrt{6} )$$, $$( 5, -2\sqrt{6} )$$, $$( -5, 2\sqrt{6} )$$ and $$( -5, -2\sqrt{6} )$$

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$$\displaystyle{ \frac{x^2}{4} + \frac{y^2}{9} = 1 }$$ and $$x^2 + y^2 = 9$$

Problem Statement

$$\displaystyle{ \frac{x^2}{4} + \frac{y^2}{9} = 1 }$$ and $$x^2 + y^2 = 9$$

$$( 0, 3 )$$ and $$( 0, -3 )$$

Problem Statement

$$\displaystyle{ \frac{x^2}{4} + \frac{y^2}{9} = 1 }$$ and $$x^2 + y^2 = 9$$

Solution

### Michel vanBiezen - 4118 video solution

video by Michel vanBiezen

$$( 0, 3 )$$ and $$( 0, -3 )$$

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$$2x^2 - y^2 = 1$$ and $$xy = -1$$

Problem Statement

$$2x^2 - y^2 = 1$$ and $$xy = -1$$

$$(1,-1)$$ and $$( -1,1 )$$

Problem Statement

$$2x^2 - y^2 = 1$$ and $$xy = -1$$

Solution

### Michel vanBiezen - 4119 video solution

video by Michel vanBiezen

$$(1,-1)$$ and $$( -1,1 )$$

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$$x^2 + 2xy - 2y^2 = 6$$ and $$-x^2 + 3xy + 2y^2 = -6$$

Problem Statement

$$x^2 + 2xy - 2y^2 = 6$$ and $$-x^2 + 3xy + 2y^2 = -6$$

$$( \sqrt{6}, 0 )$$ and $$( -\sqrt{6}, 0 )$$

Problem Statement

$$x^2 + 2xy - 2y^2 = 6$$ and $$-x^2 + 3xy + 2y^2 = -6$$

Solution

### Michel vanBiezen - 4120 video solution

video by Michel vanBiezen

$$( \sqrt{6}, 0 )$$ and $$( -\sqrt{6}, 0 )$$

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$$y=x^2-9$$ and $$y=9-x^2$$

Problem Statement

$$y=x^2-9$$ and $$y=9-x^2$$

Solution

### MIP4U - 1777 video solution

video by MIP4U

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$$x^2-5y=6 \text{ and } x^2+y=18$$

Problem Statement

$$x^2-5y=6 \text{ and } x^2+y=18$$

Solution

### MIP4U - 1778 video solution

video by MIP4U

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$$2x^2-y^2=23 \text{ and } x^2+2y^2=34$$

Problem Statement

$$2x^2-y^2=23 \text{ and } x^2+2y^2=34$$

Solution

### MIP4U - 1779 video solution

video by MIP4U

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$$2x^2-10y^2=8 \text{ and } x^2-3y^2=6$$

Problem Statement

$$2x^2-10y^2=8 \text{ and } x^2-3y^2=6$$

Solution

### Brightstorm - 1780 video solution

video by Brightstorm

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$$y=x^2 \text{ and } 3x+y=10$$

Problem Statement

$$y=x^2 \text{ and } 3x+y=10$$

Solution

### Your Math Gal - 1789 video solution

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$$x^2+y^2=25 \text{ and } 3x+4y=0$$

Problem Statement

$$x^2+y^2=25 \text{ and } 3x+4y=0$$

Solution

### Your Math Gal - 1790 video solution

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$$y-x=4 \text{ and } x^2+8x=y-16$$

Problem Statement

$$y-x=4 \text{ and } x^2+8x=y-16$$

Solution

### Your Math Gal - 1791 video solution

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$$6x-y=5 \text{ and } xy=1$$

Problem Statement

$$6x-y=5 \text{ and } xy=1$$

Solution

### Your Math Gal - 1792 video solution

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Find the solution(s) in the first quadrant to $$y = 2x^2 + 3$$ and $$y = -11x + 9$$

Problem Statement

Find the solution(s) in the first quadrant to $$y = 2x^2 + 3$$ and $$y = -11x + 9$$

$$( 0.5, 3.5 )$$

Problem Statement

Find the solution(s) in the first quadrant to $$y = 2x^2 + 3$$ and $$y = -11x + 9$$

Solution

### Michel vanBiezen - 4123 video solution

video by Michel vanBiezen

$$( 0.5, 3.5 )$$

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$$x^2 + y^2 = 25$$ and $$y = 4x/3$$

Problem Statement

$$x^2 + y^2 = 25$$ and $$y = 4x/3$$

$$( 3,4 )$$ and $$( -3,-4 )$$

Problem Statement

$$x^2 + y^2 = 25$$ and $$y = 4x/3$$

Solution

### yaymath - 4124 video solution

video by yaymath

$$( 3,4 )$$ and $$( -3,-4 )$$

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$$y = -x^2 + 4$$ and $$y = 2x + 1$$

Problem Statement

$$y = -x^2 + 4$$ and $$y = 2x + 1$$

$$( -3,-5 )$$ and $$( 1,3 )$$

Problem Statement

$$y = -x^2 + 4$$ and $$y = 2x + 1$$

Solution

### yaymath - 4125 video solution

video by yaymath

$$( -3,-5 )$$ and $$( 1,3 )$$

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$$x^2 + y^2 = 45$$ and $$y^2 - x^2 = 27$$

Problem Statement

$$x^2 + y^2 = 45$$ and $$y^2 - x^2 = 27$$

4 solutions, all combinations of $$( \pm 3, \pm 6 )$$

Problem Statement

$$x^2 + y^2 = 45$$ and $$y^2 - x^2 = 27$$

Solution

### yaymath - 4126 video solution

video by yaymath

4 solutions, all combinations of $$( \pm 3, \pm 6 )$$

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$$y=x/2 \text{ and } 2x^2-y^2=7$$

Problem Statement

$$y=x/2 \text{ and } 2x^2-y^2=7$$

Solution

### Khan Academy - 1784 video solution

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$$y=x+1$$ and $$x^2+y^2=25$$

Problem Statement

$$y=x+1$$ and $$x^2+y^2=25$$

Solution

### Khan Academy - 1785 video solution

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$$y=-x^2+6 \text{ and }y=-2x-2$$

Problem Statement

$$y=-x^2+6 \text{ and }y=-2x-2$$

Solution

### Khan Academy - 1786 video solution

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$$y=2x^2+3x-6 \text{ and } y=-x^2$$

Problem Statement

$$y=2x^2+3x-6 \text{ and } y=-x^2$$

Solution

### Khan Academy - 1787 video solution

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$$y=2(x-4)^2+3 \text{ and } y=-x^2+2x-2$$

Problem Statement

$$y=2(x-4)^2+3 \text{ and } y=-x^2+2x-2$$

Solution

### Khan Academy - 1788 video solution

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Intermediate

$$x^3 + 9x^2y = 10$$ and $$y^3 + xy^2 = 2$$

Problem Statement

$$x^3 + 9x^2y = 10$$ and $$y^3 + xy^2 = 2$$

Hint

Expand out $$(x + 3y)^3$$.

Problem Statement

$$x^3 + 9x^2y = 10$$ and $$y^3 + xy^2 = 2$$

Hint

Expand out $$(x + 3y)^3$$.

Solution

### blackpenredpen - 3784 video solution

video by blackpenredpen

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Word Problems

The area of a rectangle is $$8$$ and the perimeter is 12. What are the dimensions?

Problem Statement

The area of a rectangle is $$8$$ and the perimeter is 12. What are the dimensions?

$$( 2,4 )$$ and $$( 4,2 )$$

Problem Statement

The area of a rectangle is $$8$$ and the perimeter is 12. What are the dimensions?

Solution

### Michel vanBiezen - 4121 video solution

video by Michel vanBiezen

$$( 2,4 )$$ and $$( 4,2 )$$

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The sum of the first number squared and the second number squared equals 73. The difference between second number squared and three times the first number squared equals 37. Find all possible pairs of numbers.

Problem Statement

The sum of the first number squared and the second number squared equals 73. The difference between second number squared and three times the first number squared equals 37. Find all possible pairs of numbers.

$$( \pm 3, \pm 8 )$$

Problem Statement

The sum of the first number squared and the second number squared equals 73. The difference between second number squared and three times the first number squared equals 37. Find all possible pairs of numbers.

Solution

### Michel vanBiezen - 4122 video solution

video by Michel vanBiezen

$$( \pm 3, \pm 8 )$$

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Inequalities

$$x^2 + y^2 \leq 16$$ and $$x^2 - y^2 \gt 9$$
For this problem, do not evaluate points. Just graph and shade the area(s) that represent your answer.

Problem Statement

$$x^2 + y^2 \leq 16$$ and $$x^2 - y^2 \gt 9$$
For this problem, do not evaluate points. Just graph and shade the area(s) that represent your answer.

Solution

### yaymath - 4127 video solution

video by yaymath

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