Non Linear Systems - College Algebra

Card 0 of 6

Question

Solve the following system of equations

$\1: x^2+y^2=9\2: 3x+y=1$

Answer

First lets solve equation 2 for y.

$\3x+y=1\y=1-3x$

Now we plug in into equation 1.

FOIL the left hand side, simplify and then subtract from each side.

$\x^2+(1-3x)^2=9\x^2+1-6x+9x^2=9\10x^2-6x-8=0$

Now use the quadratic equation in order to solve for .

Recall the quadratic formula

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ , where refer to the coefficients in the quadratic equation .

$x=\frac{6\pm\sqrt{(-6)^2-4(10)(-8)}}{2(10)}$

$x=\frac{6\pm\sqrt{36+320}}{20}$

$x=\frac{6\pm\sqrt{356}}{20}$

$x=\frac{6\pm\sqrt{4\cdot 89}}{20}$

$x=\frac{6\pm2\sqrt{89}}{20}$

So the solutions for x are as follows

$x=\frac{6+2\sqrt{89}}{20}=\frac{3+\sqrt{89}}{10}$

$x=\frac{6-2\sqrt{89}}{20}=\frac{3-\sqrt{89}}{10}$

Now we just plug these values of x into equation 2 to solve for y.

$y=1-3(\frac{3+\sqrt{89}}{10})$

$y=\frac{10}{10}-(\frac{9+3\sqrt{89}}{10})=\frac{1+3\sqrt{89}}{10}$

$y=1-3(\frac{3-\sqrt{89}}{10})$

$y=\frac{10}{10}-(\frac{9-3\sqrt{89}}{10})=\frac{1-3\sqrt{89}}{10}$

So the solutions to these pair of equations are:

$x=\frac{3+\sqrt{89}}{10}$

$x=\frac{3-\sqrt{89}}{10}$

$y=\frac{1+3\sqrt{89}}{10}$

$y=\frac{1-3\sqrt{89}}{10}$

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Question

Solve the system of equations to find the points where the graphs of the equations intersect.

$\frac{1}{4}(x^2+1)=y$

Answer

Solve the system of equations,

(1)

$\frac{1}{4}(x^2+1)=y$ (2)

Notice that we can multiply both sides of equation (2) by to obtain,

(3)

Now equate equations (3) and (4) to obtain,

This is a quadratic equation, rearrange to find the roots by setting to zero and factoring,

$x = \left { -1, 2\right }$

Use these values to find the corresponding values of by substituting into either of the original equations. Using equation (1),

$y_1 = \frac{1}{4}(-1+3)=\frac{1}{2}$

$y_2 =\frac{1}{4}(2+3)=\frac{5}{4}$

$y = \left { \frac{1}{2}, \frac{5}{4}\right }$

The points $\left(-1, \frac{1}{2}\right)$ and $\left(2, \frac{5}{4} \right )$ are the intersection points of the two functions,

$y=\frac{1}{4}(x+3)$

$y=\frac{1}{4}(x^2+1)$

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Question

Solve the following system of equations:

$y=x^{2}+1$

Answer

$y=x^{2}+1$

Subtract the second equation from the first, to eliminate the y terms:

$y=x^{2}+1$

-

This yields the following: $0=x^{2}-2x$

Factor out an x from both terms:

Solve for x:

Plug in the values for x into the first equation:

$y=(0)^{2}+1=1$

$y=(2)^{2}+1=4+1=5$

Therefore, when , , and when ,

Plug in the values for x and y to make sure they satisfy the second equation too:

Solution:

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Question

Solve the following system of equations:

$x=y^{2}$

Answer

$x=y^{2}$

Substitute the value of x from the second equation, into the first equation: $y=(y^{2})-1$

Subtract y from both sides of the equation: $0=y^{2}-1-y$

Use the quadratic formula to solve for y: $y=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}=\frac{1\pm \sqrt{1-4(1)(-1)}}{2}=\frac{1\pm \sqrt{5}}{2}$

$y=\frac{1+\sqrt{5}}{2}$ or $\frac{1-\sqrt{5}}{2}$

Plug the first value of y into the first equation: $\frac{1+\sqrt{5}}{2}=x-1$

Re-write the equation to make it easier to solve for x: $\frac{1+\sqrt{5}}{2}=x-\frac{2}{2}$

Add $\frac{2}{2}$ to each side of the equation: $x=\frac{3+\sqrt{5}}{2}$

The second value of y into the first equation: $\frac{1-\sqrt{5}}{2}=x-1$

Re-write the equation to make it easier to solve for x: $\frac{1-\sqrt{5}}{2}=x-\frac{2}{2}$

Add $\frac{2}{2}$ to each side of the equation: $x=\frac{3-\sqrt{5}}{2}$ .

Plug the values of x and y into the second equation to make sure they satisfy both of them:

$(\frac{1+\sqrt{5}}{2})^{2}=\frac{1+2\sqrt{5}+5}{4}=\frac{6+2\sqrt{5}}{4}=\frac{3+\sqrt{5}}{2}$

$(\frac{1-\sqrt{5}}{2})^{2}=\frac{1-2\sqrt{5}}{4}=\frac{1-2\sqrt{5}+5}{4}=\frac{6-2\sqrt{5}}{4}=\frac{3-\sqrt{5}}{2}$

Solution: $(\frac{3+\sqrt{5}}{2},\frac{1+\sqrt{5}}{2})(\frac{3-\sqrt{5}}{2},\frac{1-\sqrt{5}}{2})$

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Question

Solve the following system of equations:

$\frac{x^{2}}{3}+\frac{y^{2}}{2}=1$

Answer

$\frac{x^{2}}{3}+\frac{y^{2}}{2}=1$

Multiply the second equation by 6, to eliminate the denominator: $6(\frac{x^{2}}{3}+\frac{y^{2}}{2}=1)6$

Simplify: $2x^{2}+3y^{2}=6$

Plug the value of y, from the first equation, into the second equation: $2x^{2}+3(x-1)^{2}=6$

Simplify: $2x^{2}+3x^{2}-6x+3=6$

Simplify further: $5x^{2}-6x-3=0$

Use the quadratic formula to solve for x: $x=\frac{6\pm \sqrt{36-4(5)(-3)}}{10}=\frac{6\pm \sqrt{96}}{10}=\frac{6\pm 4\sqrt{6}}{10}=\frac{3\pm 2\sqrt{6}}{5}$

$x=\frac{3+2\sqrt{6}}{5}$ or $x=\frac{3-2\sqrt{6}}{5}$

Plug the first value of x into the first equation: $y=\frac{3+2\sqrt{6}}{5}-1=\frac{3+2\sqrt{6}}{5}-\frac{5}{5}=\frac{2\sqrt{6}-2}{5}$

Plug the second value of x into the first equation: $y=\frac{3-2\sqrt{6}}{5}-1=\frac{3-2\sqrt{6}}{5}-\frac{5}{5}=\frac{-2\sqrt{6}-2}{5}$

Plug the values of x and y into the second equation to make sure they satisfy both of them:

$\frac{(\frac{3+2\sqrt{6}}{5})^{2}}{3}+\frac{(\frac{2\sqrt{6}-2}{5})^{2}}{2}=(\frac{9+12\sqrt{6}+24}{25})(\frac{1}{3})+(\frac{24-8\sqrt{6}+4}{5})(\frac{1}{2})=$

$\frac{11+4\sqrt{6}}{25}+\frac{14-4\sqrt{6}}{25}=\frac{25}{25}=1$

$\frac{(\frac{3-2\sqrt{6}}{5})^{2}}{3}+\frac{(\frac{-2\sqrt{6}-2}{5})^{2}}{2}=(\frac{9-12\sqrt{6}+24}{25})(\frac{1}{3})+(\frac{24+8\sqrt{6}+4}{25})(\frac{1}{2})=\frac{11-4\sqrt{6}}{25}+\frac{14+4\sqrt{6}}{25}=\frac{25}{25}=1$

Solution: $(\frac{3+2\sqrt{6}}{5},\frac{2\sqrt{6}-2}{5})(\frac{3-2\sqrt{6}}{5},\frac{-2\sqrt{6}-2}{5})$

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Question

Solve the following system of equations:

$x=2y^{2}$

Answer

$x=2y^{2}$

Substitute the value of y from the second equation, into the first equation:

$x=2y^{2}=2(-x+1)^{2}=2(x^{2}-2x+1)=2x^{2}-4x+2$

Subtract x from both sides of the equation:

$2x^{2}-5x+2=0$

Use the quadratic formula to solve for x: $x=\frac{5\pm \sqrt{25-4(2)(2)}}{4}=\frac{5\pm \sqrt{25-16}}{4}=\frac{5\pm \sqrt{9}}{4}=\frac{5\pm 3}{4}$

or $x=\frac{1}{2}$

Plug the first value of x into the second equation:

Plug the second value of x into the second equation: $y=-(\frac{1}{2})+1=\frac{1}{2}$

or $x=\frac{1}{2},y=\frac{1}{2}$

Verify the first solution by plugging the first values of x and y into both equations:

$(2)=2(-1)^{2}=2$

Verify the second solution by plugging the second values of x and y into both equations:

$(\frac{1}{2})=2(\frac{1}{2})^{2}=2(\frac{1}{4})=\frac{1}{2}$

$(\frac{1}{2})=-(\frac{1}{2})+1=\frac{1}{2}$

Solution:

$(2,-1)(\frac{1}{2},\frac{1}{2})$

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