About Nonlinear Systems of Equations:

A nonlinear system of equations is one in which at least one equation of the system is not linear. Most often, we will solve such a system using substitution, elimination, or a combination of the two.

Test Objectives
  • Demonstrate the ability to solve a nonlinear system of equations
Nonlinear Systems of Equations Practice Test:

#1:

Instructions: Solve each system.

$$a)\hspace{.2em}5x^2 + 3y^2 + 22x + 2y + 23=0$$ $$x - 2y=0$$

$$b)\hspace{.2em}{-}10x^2 + 10y^2 - 23x - 6y - 34=0$$ $$x + 2y - 2=0$$

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#2:

Instructions: Solve each system.

$$a)\hspace{.2em}x^2 + y^2 - 24x - 16=0$$ $$3x - y=-4$$

$$b)\hspace{.2em}2x^2 + y^2 - 14x - 13=0$$ $$x + y + 4=0$$

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#3:

Instructions: Solve each system.

$$a)\hspace{.2em}{-}x^2 + y^2 - 4y - 5=0$$ $$x^2 + 5y^2 + 16y + 11=0$$

$$b)\hspace{.2em}{-}x^2 + y^2 + 12x - 3y - 34=0$$ $$-x^2 + y^2 + 12x + 3y - 46=0$$

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#4:

Instructions: Solve each system.

$$a)\hspace{.2em}{-}2y^2 + x + 28y - 79=0$$ $$4y^2 + x - 56y + 161=0$$

$$b)\hspace{.2em}x^2 + y^2 - 18x - 2y + 18=0$$ $$18x^2 - y^2 - 172x + 2y + 153=0$$

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#5:

Instructions: Solve each system.

$$a)\hspace{.2em}xy=-15$$ $$4x + 3y=3$$

$$b)\hspace{.2em}2x + |y|=4$$ $$x^2 + y^2=5$$

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Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}(-2, -1)$$

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$$5x^2 + 3y^2 + 22x + 2y + 23=0$$
$$x - 2y=0$$
Graphing our system of nonlinear equations

$$b)\hspace{.2em}(-2, 2)$$

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$${-}10x^2 + 10y^2 - 23x - 6y - 34=0$$
$$x + 2y - 2=0$$
Graphing our system of nonlinear equations

Watch the Step by Step Video Solution

#2:

Solutions:

$$a)\hspace{.2em}(0, 4)$$

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$$x^2 + y^2 - 24x - 16=0$$
$$3x - y=-4$$
Graphing our system of nonlinear equations

$$b)\hspace{.2em}(1, -5)$$

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$$2x^2 + y^2 - 14x - 13=0$$
$$x + y + 4=0$$
Graphing our system of nonlinear equations

Watch the Step by Step Video Solution

#3:

Solutions:

$$a)\hspace{.2em}(0, -1)$$

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$${-}x^2 + y^2 - 4y - 5=0$$
$$x^2 + 5y^2 + 16y + 11=0$$
Graphing our system of nonlinear equations

$$b)\hspace{.2em}(6, 2)$$

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$${-}x^2 + y^2 + 12x - 3y - 34=0$$
$$-x^2 + y^2 + 12x + 3y - 46=0$$
Graphing our system of nonlinear equations

Watch the Step by Step Video Solution

#4:

Solutions:

$$a)\hspace{.2em}(-1, 10), (-1, 4)$$

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$${-}2y^2 + x + 28y - 79=0$$
$$4y^2 + x - 56y + 161=0$$
Graphing our system of nonlinear equations

$$b)\hspace{.2em}(1, 1), (9, 9), (9, -7)$$

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$$x^2 + y^2 - 18x - 2y + 18=0$$
$$18x^2 - y^2 - 172x + 2y + 153=0$$
Graphing our system of nonlinear equations

Watch the Step by Step Video Solution

#5:

Solutions:

$$a)\hspace{.2em}(-3, 5), \left(\frac{15}{4}, -4\right)$$

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$$xy=-15$$
$$4x + 3y=3$$
Graphing our system of nonlinear equations

$$b)\hspace{.2em}(1, -2), (1, 2)$$

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$$2x + |y|=4$$
$$x^2 + y^2=5$$
Graphing our system of nonlinear equations

Watch the Step by Step Video Solution