Lesson Objectives
• Demonstrate an understanding of the substitution and elimination methods
• Learn how to solve a non-linear system using substitution
• Learn how to solve a non-linear system using elimination
• Learn how to solve a non-linear system using a combination of substitution and elimination

## How to Solve a System of Non-Linear Equations

We previously learned how to solve a system of linear equations using graphing, substitution, and elimination. In this lesson, we will learn how to solve non-linear systems. An equation where some terms have two or more variables multiplied together or a variable that contains an exponent of 2 or larger is referred to as a non-linear equation. A system of non-linear equations includes two or more equations where at least one of the equations is non-linear.

### Solving a Non-Linear System by Substitution

Let's look at an example that can be solved using the substitution method.
Example 1: Solve each non-linear system. $$x^2 + y^2=4$$ $$x + y=-2$$ Let's solve our second equation for y: $$y=-x - 2$$ Now, we can plug in for y in the first equation: $$x^2 + (-x - 2)^2=4$$ $$x^2 + x^2 + 4x + 4=4$$ $$2x^2 + 4x=0$$ Solve by factoring: $$2x(x + 2)=0$$ $$2x=0$$ $$x=0$$ $$x + 2=0$$ $$x=-2$$ This tells us x can be 0 or x can be -2. We need to plug each one back in for x in one of the original equations. We can use either one. Since it is easier to work with the linear equation, let's use equation 2: $$0 + y=-2$$ $$y=-2$$ This gives us the ordered pair (0,-2) $$-2 + y=-2$$ $$y=0$$ This gives us the ordered pair (-2,0)
Our two solutions are:
x = 0, y = -2 (0,-2)
x = -2, y = 0 (-2,0)

### Solving a Non-Linear System by Elimination

When two second-degree equations are involved, we normally use the elimination method. Let's take a look at an example.
Example 2: Solve each non-linear system. $$3x^2 + 2y^2=12$$ $$x^2 + 2y^2=4$$ We can multiply the second equation by -1. This will give us opposite coefficients on y2: $$3x^2 + 2y^2=12$$ $$-x^2 - 2y^2=-4$$ Now, we can add the left sides together and set this equal to the sum of the right sides. $$\require{cancel}3x^2 + \cancel{2y^2}=12$$ $$\underline{-x^2 + \cancel{-2y^2}=-4}$$ $$2x^2=8$$ Divide each side by 2: $$x^2=4$$ Solve using the square root property: $$x=\pm \sqrt{4}$$ $$x=2,-2$$ Now, we can plug these solutions back in for x in one of the original equations: $$3(2)^2 + 2y^2=12$$ $$12 + 2y^2=12$$ $$2y^2=0$$ $$y^2=0$$ $$y=0$$ This gives us the ordered pair (2,0) $$3(-2)^2 + 2y^2=12$$ $$12 + 2y^2=12$$ $$2y^2=0$$ $$y^2=0$$ $$y=0$$ This gives us the ordered pair (-2,0)
Our two solutions are:
x = 2, y = 0 (2,0)
x = -2, y = 0 (-2,0)

### Solving a Non-Linear System using a Combination of Substitution and Elimination

Example 3: Solve each non-linear system.$$-2y^2 + 3xy + 2=0$$ $$4y^2 + 9x^2 - 10=0$$ To solve this equation, we will use a combination of the elimination and substitution methods. Let's multiply the first equation by 2. This will give us opposite coefficients (-4 and 4) on y2. $$-4y^2 + 6xy + 4=0$$ $$4y^2 + 9x^2 - 10=0$$ Let's move our constants to the right side: $$-4y^2 + 6xy=-4$$ $$4y^2 + 9x^2=10$$ Now, we will add the left sides and set this equal to the sum of the right sides: $$\cancel{-4y^2}+ 6xy=-4$$ $$\underline{\cancel{4y^2}+ 9x^2=10}$$ $$9x^2 + 6xy=6$$ We can divide each side by 3: $$3x^2 + 2xy=2$$ Now, we will solve the equation for y: $$2xy=-3x^2 + 2$$ $$y=\frac{-3x^2 + 2}{2x}$$ Now, we will substitute in for y in one of the original equations. Using the second equation will be a bit easier. $$4y^2 + 9x^2 - 10=0$$ $$4\left(\frac{-3x^2 + 2}{2x}\right)^2 + 9x^2 - 10=0$$ $$4\frac{(-3x^2 + 2)^2}{4x^2}+ 9x^2 - 10=0$$ $$\cancel{4}\frac{(9x^4-12x^2+4)}{\cancel{4}x^2}+ 9x^2 - 10=0$$ $$\frac{(9x^4-12x^2+4)}{x^2}+ 9x^2 - 10=0$$ We can multiply each side by x2 to clear the denominator: $$9x^4-12x^2+4 + 9x^4 - 10x^2=0$$ $$18x^4-22x^2+4=0$$ We can divide each side of the equation by 2: $$9x^4 - 11x^2 + 2=0$$ This equation is quadratic in form and can be solved by factoring: $$(x + 1)(x - 1)(9x^2 - 2)=0$$ $$x + 1=0$$ $$x=-1$$ $$x - 1=0$$ $$x=1$$ $$9x^2 - 2=0$$ $$9x^2=2$$ $$x^2=\frac{2}{9}$$ $$x=\pm \sqrt{\frac{2}{9}}$$ $$x=\pm \frac{\sqrt{2}}{3}$$ So we have four values for x: $$x=\pm 1, \pm\frac{\sqrt{2}}{3}$$ To find y, we will use our equation: $$y=\frac{-3x^2 + 2}{2x}$$ If x is 1: $$y=\frac{-3(1)^2 + 2}{2(1)}$$ $$y=\frac{-3 + 2}{2}$$ $$y=\frac{-1}{2}$$ $$x=1, y=\frac{-1}{2}$$ $$\left(1,\frac{-1}{2}\right)$$ If x is -1: $$y=\frac{-3(-1)^2 + 2}{2(-1)}$$ $$y=\frac{-3 + 2}{-2}$$ $$y=\frac{-1}{-2}$$ $$x=-1, y=\frac{1}{2}$$ $$\left(-1,\frac{1}{2}\right)$$ If x is square root of 2 over 3: $$y=\frac{-3\left(\frac{\sqrt{2}}{3}\right)^2 + 2}{2\left(\frac{\sqrt{2}}{3}\right)}$$ $$y=\frac{-3\left(\frac{2}{9}\right) + 2}{\frac{2\sqrt{2}}{3}}$$ $$y=\large{\frac{-\frac{2}{3}+ 2}{\frac{2\sqrt{2}}{3}}}$$ $$y=\large{\frac{\frac{4}{3}}{\frac{2\sqrt{2}}{3}}}$$ $$y=\frac{2\cancel{4}}{\cancel{3}}\cdot \frac{\cancel{3}}{\cancel{2}\sqrt{2}}$$ $$y=\frac{2}{\sqrt{2}}$$ If we rationalize the denominator we end up with: $$y=\sqrt{2}$$ $$x=\frac{\sqrt{2}}{3}, y=\sqrt{2}$$ $$\left(\frac{\sqrt{2}}{3},\sqrt{2}\right)$$ If x is the negative square root of 2 over 3: $$y=\frac{-3\left(\frac{-\sqrt{2}}{3}\right)^2 + 2}{2\left(\frac{-\sqrt{2}}{3}\right)}$$ $$y=\frac{-3\left(\frac{2}{9}\right) + 2}{\frac{-2\sqrt{2}}{3}}$$ $$y=\large{\frac{-\frac{2}{3}+ 2}{\frac{-2\sqrt{2}}{3}}}$$ $$y=\large{\frac{\frac{4}{3}}{\frac{-2\sqrt{2}}{3}}}$$ $$y=\frac{2\cancel{4}}{\cancel{3}}\cdot \frac{\cancel{3}}{-1\cancel{2}\sqrt{2}}$$ $$y=\frac{2}{-\sqrt{2}}$$ If we rationalize the denominator we end up with: $$y=-\sqrt{2}$$ $$x=-\frac{\sqrt{2}}{3}, y=-\sqrt{2}$$ $$\left(-\frac{\sqrt{2}}{3},-\sqrt{2}\right)$$

#### Skills Check:

Example #1

Solve each equation. $$2x^{2}+ 4y^{2}+ 3x + 66y=-72$$ $$x + 2y=0$$

A
$$(4, 2)$$
B
$$(6, -3), (4, 2)$$
C
$$(1, 4), (6, -3)$$
D
$$(-2, 5), (7, -9)$$
E
$$(6, -3), (4, -2)$$

Example #2

Solve each equation. $$x^{2}+ 2x + y=-7$$ $$y=7x^{2}+ 14x - 7$$

A
$$(-2, -7), (9, 1)$$
B
$$(-2, -7), (0, -7)$$
C
$$(1, 3), (-4, -5)$$
D
$$\text{No Solution}$$
E
$$(4, 9), (6, 10)$$

Example #3

Solve each equation. $$x^{2}+ y^{2}- 11x + 6y=66$$ $$x^{2}+ y^{2}- 29x + 6y=102$$

A
$$(-2, 4), (-2, -10)$$
B
$$(-2, 1), (3, 5)$$
C
$$(8, 7), (5, -4)$$
D
$$(3, -2), (1, 5)$$
E
$$(-2, 1), (-6, -4)$$

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