Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

How many solutions does the system of equations below have?\newline3x+3y+2z=153x + 3y + 2z = 15\newline−2x−2y+z=11-2x - 2y + z = 11\newline2x−y+z=42x - y + z = 4\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Precalculusright chevron icon
Determine the number of solutions to a system of equations in three variablesright chevron icon

Full solution

Q. How many solutions does the system of equations below have?\newline3x+3y+2z=153x + 3y + 2z = 15\newline−2x−2y+z=11-2x - 2y + z = 11\newline2x−y+z=42x - y + z = 4\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Combine Equations to Eliminate zz: First, let's try to simplify the system by adding the first and second equations to eliminate zz.(3x+3y+2z)+(−2x−2y+z)=15+11(3x + 3y + 2z) + (-2x - 2y + z) = 15 + 11This simplifies to x+y+3z=26x + y + 3z = 26
  2. Combine Equations to Eliminate z Again: Now, let's add the second and third equations to eliminate z again.\newline(−2x−2y+z)+(2x−y+z)=11+4(-2x - 2y + z) + (2x - y + z) = 11 + 4\newlineThis simplifies to −3y+2z=15-3y + 2z = 15
  3. Create New Equations: We now have two new equations:\newlinex+y+3z=26x + y + 3z = 26\newline−3y+2z=15-3y + 2z = 15\newlineLet's multiply the second equation by 33 to align the yy terms with the first equation.\newline3(−3y+2z)=3(15)3(-3y + 2z) = 3(15)\newlineThis gives us −9y+6z=45-9y + 6z = 45
  4. Align y Terms: Now we can add the new equation to the first one to eliminate y.\newline(x+y+3z)+(−9y+6z)=26+45(x + y + 3z) + (-9y + 6z) = 26 + 45\newlineThis simplifies to x−8y+9z=71x - 8y + 9z = 71

More problems from Determine the number of solutions to a system of equations in three variables

Question
Solve using substitution. 5x−2y=−7 5x - 2y = -7 x=−5 x = -5 (_,_)
Get tutor helpright-arrow

Posted 2 months ago

Question
Is (1,1)(1,1) a solution to this system of equations? \newline4x+10y=144x + 10y = 14\newlinex+6y=7x + 6y = 7\newlineChoices:\newline(A) yes \newline(B) no
Get tutor helpright-arrow

Posted 2 months ago

Question
Which describes the system of equations below?\newliney=–3x+9y = –3x + 9 \newliney=–3x+9y = –3x + 9 \newlineChoices:\newline(A) consistent and independent\text{(A) consistent and independent} \newline(B) consistent and dependent\text{(B) consistent and dependent} \newline(C) inconsistent\text{(C) inconsistent}
Get tutor helpright-arrow

Posted 2 months ago

Question
Solve using elimination. \newline7x−8y=−17 7x - 8y = -17 \newline −7x+3y=2 -7x + 3y = 2 \newline (____,____)(\_\_\_\_, \_\_\_\_)
Get tutor helpright-arrow

Posted 2 months ago

Question
Solve. \newlinex=−2 x = -2 \newline−2x+2y=−8 -2x + 2y = -8 \newline(____,____)(\_\_\_\_, \_\_\_\_)
Get tutor helpright-arrow

Posted 2 months ago

Question
Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineAt a community barbecue, Mrs. Wilkerson and Mr. Hogan are buying dinner for their families. Mrs. Wilkerson purchases 33 hot dog meals and 33 hamburger meals, paying a total of $36\$36. Mr. Hogan buys 11 hot dog meal and 33 hamburger meals, spending $26\$26 in all. How much do the meals cost?\newlineHot dog meals cost $\$_______ each, and hamburger meals cost $\$________ each.
Get tutor helpright-arrow

Posted 2 months ago

Question
Solve the system of equations by substitution.\newline−3x−y−3z=−11-3x - y - 3z = -11\newlinez=5z = 5\newlinex−y+3z=19x - y + 3z = 19\newline(____.____,____)
Get tutor helpright-arrow

Posted 2 months ago

Question
Solve the system of equations by elimination.\newlinex−3y−2z=10 x - 3y - 2z = 10 \newline3x+2y+2z=14 3x + 2y + 2z = 14 \newline2x−3y−2z=16 2x - 3y - 2z = 16 \newline(_,_,_)(\_,\_,\_)
Get tutor helpright-arrow

Posted 1 month ago

Question
Solve the system of equations.\newliney=x2+36x+3 y = x^2 + 36x + 3 \newliney=22x−37 y = 22x - 37 \newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(_,_)(\_,\_)\newline(_,_)(\_,\_)
Get tutor helpright-arrow

Posted 1 month ago

Question
Solve the system of equations.\newliney=−x−24 y = -x - 24 \newlinex2+y2=488 x^2 + y^2 = 488 \newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(_,_) (\_,\_) \newline(_,_) (\_,\_)
Get tutor helpright-arrow

Posted 2 months ago

View all (29)