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# 11.6E: Exercises

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• OpenStax
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### Practice Makes Perfect

##### Exercise $$\PageIndex{17}$$ Solve a System of Nonlinear Equations Using Graphing

In the following exercises, solve the system of equations by using graphing.

1. $$\left\{\begin{array}{l}{y=2 x+2} \\ {y=-x^{2}+2}\end{array}\right.$$
2. $$\left\{\begin{array}{l}{y=6 x-4} \\ {y=2 x^{2}}\end{array}\right.$$
3. $$\left\{\begin{array}{l}{x+y=2} \\ {x=y^{2}}\end{array}\right.$$
4. $$\left\{\begin{array}{l}{x-y=-2} \\ {x=y^{2}}\end{array}\right.$$
5. $$\left\{\begin{array}{l}{y=\frac{3}{2} x+3} \\ {y=-x^{2}+2}\end{array}\right.$$
6. $$\left\{\begin{array}{l}{y=x-1} \\ {y=x^{2}+1}\end{array}\right.$$
7. $$\left\{\begin{array}{l}{x=-2} \\ {x^{2}+y^{2}=4}\end{array}\right.$$
8. $$\left\{\begin{array}{l}{y=-4} \\ {x^{2}+y^{2}=16}\end{array}\right.$$
9. $$\left\{\begin{array}{l}{x=2} \\ {(x+2)^{2}+(y+3)^{2}=16}\end{array}\right.$$
10. $$\left\{\begin{array}{l}{y=-1} \\ {(x-2)^{2}+(y-4)^{2}=25}\end{array}\right.$$
11. $$\left\{\begin{array}{l}{y=-2 x+4} \\ {y=\sqrt{x}+1}\end{array}\right.$$
12. $$\left\{\begin{array}{l}{y=-\frac{1}{2} x+2} \\ {y=\sqrt{x}-2}\end{array}\right.$$

2.

4.

6.

8.

10.

12.

##### Exercise $$\PageIndex{18}$$ Solve a System of Nonlinear Equations Using Substitution

In the following exercises, solve the system of equations by using substitution.

1. $$\left\{\begin{array}{l}{x^{2}+4 y^{2}=4} \\ {y=\frac{1}{2} x-1}\end{array}\right.$$
2. $$\left\{\begin{array}{l}{9 x^{2}+y^{2}=9} \\ {y=3 x+3}\end{array}\right.$$
3. $$\left\{\begin{array}{l}{9 x^{2}+y^{2}=9} \\ {y=x+3}\end{array}\right.$$
4. $$\left\{\begin{array}{l}{9 x^{2}+4 y^{2}=36} \\ {x=2}\end{array}\right.$$
5. $$\left\{\begin{array}{l}{4 x^{2}+y^{2}=4} \\ {y=4}\end{array}\right.$$
6. $$\left\{\begin{array}{l}{x^{2}+y^{2}=169} \\ {x=12}\end{array}\right.$$
7. $$\left\{\begin{array}{l}{3 x^{2}-y=0} \\ {y=2 x-1}\end{array}\right.$$
8. $$\left\{\begin{array}{l}{2 y^{2}-x=0} \\ {y=x+1}\end{array}\right.$$
9. $$\left\{\begin{array}{l}{y=x^{2}+3} \\ {y=x+3}\end{array}\right.$$
10. $$\left\{\begin{array}{l}{y=x^{2}-4} \\ {y=x-4}\end{array}\right.$$
11. $$\left\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {x-y=1}\end{array}\right.$$
12. $$\left\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {2 x+y=10}\end{array}\right.$$

2. $$(-1,0),(0,3)$$

4. $$(2,0)$$

6. $$(12,-5),(12,5)$$

8. No solution

10. $$(0,-4),(1,-3)$$

12. $$(3,4),(5,0)$$

##### Exercise $$\PageIndex{19}$$ Solve a System of Nonlinear Equations Using Elimination

In the following exercises, solve the system of equations by using elimination.

1. $$\left\{\begin{array}{l}{x^{2}+y^{2}=16} \\ {x^{2}-2 y=8}\end{array}\right.$$
2. $$\left\{\begin{array}{l}{x^{2}+y^{2}=16} \\ {x^{2}-y=4}\end{array}\right.$$
3. $$\left\{\begin{array}{l}{x^{2}+y^{2}=4} \\ {x^{2}+2 y=1}\end{array}\right.$$
4. $$\left\{\begin{array}{l}{x^{2}+y^{2}=4} \\ {x^{2}-y=2}\end{array}\right.$$
5. $$\left\{\begin{array}{l}{x^{2}+y^{2}=9} \\ {x^{2}-y=3}\end{array}\right.$$
6. $$\left\{\begin{array}{l}{x^{2}+y^{2}=4} \\ {y^{2}-x=2}\end{array}\right.$$
7. $$\left\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {2 x^{2}-3 y^{2}=5}\end{array}\right.$$
8. $$\left\{\begin{array}{l}{x^{2}+y^{2}=20} \\ {x^{2}-y^{2}=-12}\end{array}\right.$$
9. $$\left\{\begin{array}{l}{x^{2}+y^{2}=13} \\ {x^{2}-y^{2}=5}\end{array}\right.$$
10. $$\left\{\begin{array}{l}{x^{2}+y^{2}=16} \\ {x^{2}-y^{2}=16}\end{array}\right.$$
11. $$\left\{\begin{array}{l}{4 x^{2}+9 y^{2}=36} \\ {2 x^{2}-9 y^{2}=18}\end{array}\right.$$
12. $$\left\{\begin{array}{l}{x^{2}-y^{2}=3} \\ {2 x^{2}+y^{2}=6}\end{array}\right.$$
13. $$\left\{\begin{array}{l}{4 x^{2}-y^{2}=4} \\ {4 x^{2}+y^{2}=4}\end{array}\right.$$
14. $$\left\{\begin{array}{l}{x^{2}-y^{2}=-5} \\ {3 x^{2}+2 y^{2}=30}\end{array}\right.$$
15. $$\left\{\begin{array}{l}{x^{2}-y^{2}=1} \\ {x^{2}-2 y=4}\end{array}\right.$$
16. $$\left\{\begin{array}{l}{2 x^{2}+y^{2}=11} \\ {x^{2}+3 y^{2}=28}\end{array}\right.$$

2. $$(0,-4),(-\sqrt{7}, 3),(\sqrt{7}, 3)$$

4. $$(0,-2),(-\sqrt{3}, 1),(\sqrt{3}, 1)$$

6. $$(-2,0),(1,-\sqrt{3}),(1, \sqrt{3})$$

8. $$(-2,-4),(-2,4),(2,-4),(2,4)$$

10. $$(-4,0),(4,0)$$

12. $$(-\sqrt{3}, 0),(\sqrt{3}, 0)$$

14. $$(-2,-3),(-2,3),(2,-3),(2,3)$$

16. $$(-1,-3),(-1,3),(1,-3),(1,3)$$

##### Exercise $$\PageIndex{20}$$ Use a System of Nonlinear Equations to Solve Applications

In the following exercises, solve the problem using a system of equations.

1. The sum of two numbers is $$−6$$ and the product is $$8$$. Find the numbers.
2. The sum of two numbers is $$11$$ and the product is $$−42$$. Find the numbers.
3. The sum of the squares of two numbers is $$65$$. The difference of the number is $$3$$. Find the numbers.
4. The sum of the squares of two numbers is $$113$$. The difference of the number is $$1$$. Find the numbers.
5. The difference of the squares of two numbers is $$15$$. The difference of twice the square of the first number and the square of the second number is $$30$$. Find the numbers.
6. The difference of the squares of two numbers is $$20$$. The difference of the square of the first number and twice the square of the second number is $$4$$. Find the numbers.
7. The perimeter of a rectangle is $$32$$ inches and its area is $$63$$ square inches. Find the length and width of the rectangle.
8. The perimeter of a rectangle is $$52$$ cm and its area is $$165$$ $$\mathrm{cm}^{2}$$. Find the length and width of the rectangle.
9. Dion purchased a new microwave. The diagonal of the door measures $$17$$ inches. The door also has an area of $$120$$ square inches. What are the length and width of the microwave door?
10. Jules purchased a microwave for his kitchen. The diagonal of the front of the microwave measures $$26$$ inches. The front also has an area of $$240$$ square inches. What are the length and width of the microwave?
11. Roman found a widescreen TV on sale, but isn’t sure if it will fit his entertainment center. The TV is $$60$$”. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of $$1728$$ square inches. His entertainment center has an insert for the TV with a length of $$50$$ inches and width of $$40$$ inches. What are the length and width of the TV screen and will it fit Roman’s entertainment center?
12. Donnette found a widescreen TV at a garage sale, but isn’t sure if it will fit her entertainment center. The TV is $$50$$”. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of $$1200$$ square inches. Her entertainment center has an insert for the TV with a length of $$38$$ inches and width of $$27$$ inches. What are the length and width of the TV screen and will it fit Donnette’s entertainment center?

2. $$-3$$ and $$14$$

4. $$-7$$ and $$-8$$ or $$8$$ and $$7$$

6. $$-6$$ and $$-4$$ or $$-6$$ and $$4$$ or $$6$$ and $$-4$$ or $$6$$ and $$4$$

8. If the length is $$11$$ cm, the width is $$15$$ cm. If the length is $$15$$ cm, the width is $$11$$ cm.

10. If the length is $$10$$ inches, the width is $$24$$ inches. If the length is $$24$$ inches, the width is $$10$$ inches.

12. The length is $$40$$ inches and the width is $$30$$ inches. The TV will not fit Donnette’s entertainment center.

##### Exercise $$\PageIndex{21}$$ Writing Exercises
1. In your own words, explain the advantages and disadvantages of solving a system of equations by graphing.
2. Explain in your own words how to solve a system of equations using substitution.
3. Explain in your own words how to solve a system of equations using elimination.
4. A circle and a parabola can intersect in ways that would result in $$0, 1, 2, 3,$$ or $$4$$ solutions. Draw a sketch of each of the possibilities.