Call Now to Set Up Tutoring:

(888) 888-0446

Algebra 1 : Systems of Equations

Study concepts, example questions & explanations for Algebra 1

Create An Account

All Algebra 1 Resources

10 Diagnostic Tests 557 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Find A Solution Set

Solve for :

$\sqrt{x} + \sqrt{x -5} = 5$

Possible Answers:

x=9

$x = -4 \textrm{ or }x = 9$

The equation has no solution.

$x = 4 \textrm{ or }x = 9$

x = -4

Correct answer:

x=9

Explanation:

Move one radical to the other side, then square, thereby yielding an equation with only one radical.

$\sqrt{x} + \sqrt{x -5} = 5$

$\sqrt{x} + \sqrt{x -5} -\sqrt{x} = 5-\sqrt{x}$

$\sqrt{x -5} = 5-\sqrt{x}$

$\left (\sqrt{x -5} \right ) ^{2} =\left ( 5-\sqrt{x }\right ) ^{2}$

$x -5= 5 ^{2} - 2 \cdot 5 \cdot \sqrt{x } + \left ( \sqrt{x }\right ) ^{2}$

$x -5= 25 - 10 \sqrt{x } + x$

Isolate the radical on one side, then square.

$x -5-x -25= 25 - 10 \sqrt{x } + x -x -25$

$-30 = - 10 \sqrt{x }$

$-30 \div (-10) = - 10 \sqrt{x } \div (-10)$

$3= \sqrt{x }$

$3^{2} =\left ( \sqrt{x } \right )^{2}$

9 = x

Substitution confirms this to be the only solution.

Report an Error

Example Question #2 : How To Find A Solution Set

If the area of a rectangle is 100 square feet and the width is 20 feet, then what is the perimeter?

Possible Answers:

30 feet

80 feet

20 feet

50 feet

Correct answer:

50 feet

Explanation:

The area of a rectangle is A=L*W , where A is the area, L is the length, and W is the width. The perimeter is given by P=2L+2W . We know that A=100 and W=20 . We can solve for L using $L=\frac{A}{W}=\frac{100}{20}=5.$ The perimeter is then 2*5+2*20=50 feet.

Report an Error

Example Question #3 : How To Find A Solution Set

Solve for :

$\sqrt[3]{2n-8} = -4$

Possible Answers:

n = -28

n = -36

The equation has no solution.

n = 32

$n = -28\textrm{ or }n = -36$

Correct answer:

n = -28

Explanation:

Cube both sides of the equation to form a linear equation, then solve:

$\sqrt[3]{2n-8} = -4$

$\dpi{150} \left (\sqrt[3]{2n-8} \right ) ^{3}= \left (-4 \right )^{3}$

2n-8 = -64

2n-8 + 8 = -64 + 8

2n = -56

$2n \div 2 = -56\div 2$

n = -28

Report an Error

Example Question #1 : Solving Equations

Solve this system of equations.

7x+4y+9z=0

8x-3y-6z=12

2x-y-3z=5

Possible Answers:

x=1 , y=2 , $z=-\frac{5}{3}$

x=5 , y=-9 , z=3

x=-2 , y=-2 , z=2

$x=\frac{1}{2}$ , $y=\frac{3}{2}$ , z=-3

x=-1 , y=2 , $z=-\frac{5}{3}$

Correct answer:

x=1 , y=2 , $z=-\frac{5}{3}$

Explanation:

Equation 1: 7x+4y+9z=0

Equation 2: 8x-3y-6z=12

Equation 3: 2x-y-3z=5

Adding the terms of the first and second equations together will yield 15x+y+3z=12 .

Then, add that to the third equation so that the y and z terms are eliminated. You will get 17x=17 .

This tells us that x = 1. Plug this x = 1 back into the systems of equations.

7+4y+9z=0

8-3y-6z=12

2-y-3z=5

Now, we can do the rest of the problem by using the substitution method. We'll take the third equation and use it to solve for y.

2-y-3z=5

-y=3z+5-2

y=-3z-3

Plug this y-equation into the first equation (or second equation; it doesn't matter) to solve for z.

7+4y+9z=0

7+4(-3z-3) +9z=0

7-12z-12+9z=0

-5-3z=0

$z=-\frac{5}{3}$

We can use this z value to find y

y=-3z-3

$y=-3\left ( -\frac{5}{3}\right )-3$

y=2

So the solution set is x = 1, y = 2, and z = –5/3.

Report an Error

Example Question #3 : How To Find A Solution Set

Solve for : $x^{2}-144=0$

Possible Answers:

x=12,-12

x=10,-14

x=14,-14

x=8,-4

Correct answer:

x=12,-12

Explanation:

To solve this problem we can first add 144 to each side of the equation yielding $x^{2}=144$

Then we take the square root of both sides to get $\sqrt{x^{2}}=\sqrt{144}$

Then we calculate the square root of 144 which is $x=\pm12$ .

Report an Error

Example Question #1 : How To Find A Solution Set

Determine where the graphs of the following equations will intersect.

3x+2y=-1

-x-3y=5

Possible Answers:

(-8,1)

(2,-3)

(-2,1)

(1,-2)

Correct answer:

(1,-2)

Explanation:

We can solve the system of equations using the substitution method.

3x+2y=-1

-x-3y=5

Solve for in the second equation.

-x-3y=5

-3y=5+x

-3y-5=x

Substitute this value of into the first equation.

3(-3y-5)+2y=-1

Now we can solve for .

-9y-15+2y=-1

-7y-15=-1

-7y=14

$y=\frac{14}{-7}=-2$

Solve for using the first equation with this new value of .

3x+2(-2)=-1

3x-4=-1

3x=3

$x=\frac{3}{3}=1$

The solution is the ordered pair (1,-2) .

Report an Error

Example Question #1 : Solving Equations And Inequallities

Give all real solutions of the following equation:

$x^{4} +5x^{2} - 36 = 0$

Possible Answers:

$\left \{ 2,3\right \}$

$\left \{ -3,3\right \}$

$\left \{ -2,2\right \}$

$\left \{ -3, -2\right \}$

$\left \{ -3, -2, 2, 3\right \}$

Correct answer:

$\left \{ -2,2\right \}$

Explanation:

By substituting $u = x^{2}$ - and, subsequently, $u^{2} =\left ( x^{2} \right )^{2} = x^{4}$ this can be rewritten as a quadratic equation, and solved as such:

$x^{4} +5x^{2} - 36 = 0$

$u^{2} + 5u - 36 = 0$

We are looking to factor the quadratic expression as (u+?)(u+?) , replacing the two question marks with integers with product and sum 5; these integers are 9,-4 .

(u+9)(u-4) = 0

Substitute back:

$(x^{2}+9)(x^{2}-4) = 0$

The first factor cannot be factored further. The second factor, however, can itself be factored as the difference of squares:

$(x^{2}+9)(x+2)(x-2) = 0$

Set each factor to zero and solve:

$x^{2}+9= 0$

$x^{2}= -9$

Since no real number squared is equal to a negative number, no real solution presents itself here.

$x-2= 0 \Rightarrow x = 2$

$x+2 = 0 \Rightarrow x = - 2$

The solution set is $\left \{ -2, 2\right \}$ .

Report an Error

Example Question #4 : How To Find A Solution Set

Solve for :

-3x + 12 = -3

Possible Answers:

Correct answer:

Explanation:

Subtract 12 from both sides:

–3x = –15

Divide both sides by –3:

x = 5

Report an Error

Example Question #3 : Solving Equations

Solve for :

x(x - 2) = x^2 -8

Possible Answers:

Correct answer:

Explanation:

Distribute the x through the parentheses:

x² –2x = x² – 8

Subtract x² from both sides:

–2x = –8

Divide both sides by –2:

x = 4

Report an Error

Example Question #5 : How To Find A Solution Set

Solve for :

$\dpi{130} \sqrt[3]{100-x} - 5 = 0$

Possible Answers:

The equation has no solution.

$x = -25 \textrm{ or } x = 225$

x = 125

x = -25

x = 225

Correct answer:

x = -25

Explanation:

Add 5 to both sides to isolate the cube root:

$\dpi{130} \sqrt[3]{100-x} - 5+ 5 = 0+ 5$

$\dpi{130} \sqrt[3]{100-x} = 5$

Cube both sides:

$\dpi{130} \left (\sqrt[3]{100-x} \right) ^{3}= 5 ^{3}$

100-x = 125

To isolate , move it to the right side of the equation. We choose the right side instead of the left side so as to make positive:

100-x +x= 125+x

100 = 125+x

Subtract 125 from both sides to isolate :

100 - 125 = 125 - 125 +x

-25 = x

Report an Error

View Algebra Tutors

Amelia
Certified Tutor

Austin College, Bachelor in Arts, Applied Mathematics. University of North Carolina at Charlotte, Master of Science, Applied ...

View Algebra Tutors

Teresa
Certified Tutor

Great Basin College, Bachelor in Arts, Social Science Teacher Education. American College of Education, Masters in Education,...

View Algebra Tutors

Soung
Certified Tutor

University of Chicago, Bachelor in Arts, Chemistry. Northwestern University, Doctor of Science, Molecular Pharmacology.

All Algebra 1 Resources

10 Diagnostic Tests 557 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Reading Tutors in San Francisco-Bay Area, LSAT Tutors in New York City, SAT Tutors in Phoenix, Math Tutors in Dallas Fort Worth, SSAT Tutors in Boston, Computer Science Tutors in Houston, Chemistry Tutors in Washington DC, Calculus Tutors in Philadelphia, Algebra Tutors in Los Angeles, Chemistry Tutors in Chicago

Popular Courses & Classes

ISEE Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Boston, GRE Courses & Classes in Chicago, GMAT Courses & Classes in Miami, Spanish Courses & Classes in New York City, ACT Courses & Classes in Denver, GRE Courses & Classes in Miami, GMAT Courses & Classes in New York City, Spanish Courses & Classes in Washington DC, ISEE Courses & Classes in Atlanta

Popular Test Prep

ACT Test Prep in Boston, GRE Test Prep in Philadelphia, ISEE Test Prep in Los Angeles, MCAT Test Prep in Dallas Fort Worth, ACT Test Prep in Dallas Fort Worth, LSAT Test Prep in Miami, LSAT Test Prep in Los Angeles, GMAT Test Prep in New York City, SAT Test Prep in Dallas Fort Worth, ACT Test Prep in Atlanta

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Algebra 1 : Systems of Equations

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : How To Find A Solution Set

Example Question #2 : How To Find A Solution Set

Example Question #3 : How To Find A Solution Set

Example Question #1 : Solving Equations

Example Question #3 : How To Find A Solution Set

Example Question #1 : How To Find A Solution Set

Example Question #1 : Solving Equations And Inequallities

Example Question #4 : How To Find A Solution Set

Example Question #3 : Solving Equations

Example Question #5 : How To Find A Solution Set

All Algebra 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: