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?\newline−2x−2y−z=3-2x - 2y - z = 3\newlinex−3y+z=17x - 3y + z = 17\newline3x+2y−z=−133x + 2y - z = -13\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
Algebra 2right 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?\newline−2x−2y−z=3-2x - 2y - z = 3\newlinex−3y+z=17x - 3y + z = 17\newline3x+2y−z=−133x + 2y - z = -13\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Eliminate z: First, let's add the first and second equations to eliminate z.\newline(−2x−2y−z)+(x−3y+z)=3+17(-2x - 2y - z) + (x - 3y + z) = 3 + 17\newline−2x−2y−z+x−3y+z=20-2x - 2y - z + x - 3y + z = 20\newline−2x+x−2y−3y=20-2x + x - 2y - 3y = 20\newline−x−5y=20-x - 5y = 20
  2. Eliminate z again: Now, let's multiply the second equation by 22 and add it to the third equation to also eliminate z.\newline2(x−3y+z)+(3x+2y−z)=2(17)+(−13)2(x - 3y + z) + (3x + 2y - z) = 2(17) + (-13)\newline2x−6y+2z+3x+2y−z=34−132x - 6y + 2z + 3x + 2y - z = 34 - 13\newline2x+3x−6y+2y+2z−z=212x + 3x - 6y + 2y + 2z - z = 21\newline5x−4y+z=215x - 4y + z = 21
  3. Eliminate y: We can now add the modified first equation to the modified third equation to eliminate y.\newline(−x−5y)+(5x−4y)=20+21(-x - 5y) + (5x - 4y) = 20 + 21\newline−x+5x−5y−4y=41-x + 5x - 5y - 4y = 41\newline4x−9y=414x - 9y = 41
  4. Solve for x: Let's solve for x using the equation 4x−9y=414x - 9y = 41. Since we don't have a value for yy, we can't solve for xx directly. We need to find another equation that relates xx and yy to solve the system.
  5. Find xx and yy: We can use the modified first equation −x−5y=20-x - 5y = 20 and the modified third equation 5x−4y+z=215x - 4y + z = 21 to solve for xx and yy. Let's multiply the first equation by 55 to match the coefficient of xx in the third equation. 5(−x−5y)=5(20)5(-x - 5y) = 5(20) −5x−25y=100-5x - 25y = 100
  6. Eliminate x: Now, let's add this new equation to the third equation to eliminate x.\newline(−5x−25y)+(5x−4y+z)=100+21(-5x - 25y) + (5x - 4y + z) = 100 + 21\newline−5x+5x−25y−4y+z=121-5x + 5x - 25y - 4y + z = 121\newline−29y+z=121-29y + z = 121
  7. Solve for yy: We now have two equations with two variables:\newline4x−9y=414x - 9y = 41\newline−29y+z=121-29y + z = 121\newlineWe can't solve for a specific value of yy or zz without another equation relating them. We need to go back and check our steps to see if we can find another relationship.
  8. Find x: Looking back at the original equations, we can multiply the second equation by 22 and add it to the first to eliminate z.\newline2(x−3y+z)+(−2x−2y−z)=2(17)+32(x - 3y + z) + (-2x - 2y - z) = 2(17) + 3\newline2x−6y+2z−2x−2y−z=34+32x - 6y + 2z - 2x - 2y - z = 34 + 3\newline−6y−2y+2z−z=37-6y - 2y + 2z - z = 37\newline−8y+z=37-8y + z = 37
  9. Find zz: We now have three equations with two variables:\newline4x−9y=414x - 9y = 41\newline−29y+z=121-29y + z = 121\newline−8y+z=37-8y + z = 37\newlineWe can use the last two equations to solve for yy and zz.
  10. Find zz: We now have three equations with two variables:\newline4x−9y=414x - 9y = 41\newline−29y+z=121-29y + z = 121\newline−8y+z=37-8y + z = 37\newlineWe can use the last two equations to solve for yy and zz.Subtract the third equation from the second to eliminate zz.\newline(−29y+z)−(−8y+z)=121−37(-29y + z) - (-8y + z) = 121 - 37\newline−29y+z+8y−z=84-29y + z + 8y - z = 84\newline−21y=84-21y = 84\newliney=−4y = -4
  11. Find zz: We now have three equations with two variables:\newline4x−9y=414x - 9y = 41\newline−29y+z=121-29y + z = 121\newline−8y+z=37-8y + z = 37\newlineWe can use the last two equations to solve for yy and zz.Subtract the third equation from the second to eliminate zz.\newline(−29y+z)−(−8y+z)=121−37(-29y + z) - (-8y + z) = 121 - 37\newline−29y+z+8y−z=84-29y + z + 8y - z = 84\newline−21y=84-21y = 84\newliney=−4y = -4Now that we have yy, we can substitute it back into the equation 4x−9y=414x - 9y = 41 to find xx.\newline4x−9(−4)=414x - 9(-4) = 41\newline4x+36=414x + 36 = 41\newline4x=41−364x = 41 - 36\newline−29y+z=121-29y + z = 12100\newline−29y+z=121-29y + z = 12111
  12. Find z: We now have three equations with two variables:\newline4x−9y=414x - 9y = 41\newline−29y+z=121-29y + z = 121\newline−8y+z=37-8y + z = 37\newlineWe can use the last two equations to solve for y and z.Subtract the third equation from the second to eliminate z.\newline(−29y+z)−(−8y+z)=121−37(-29y + z) - (-8y + z) = 121 - 37\newline−29y+z+8y−z=84-29y + z + 8y - z = 84\newline−21y=84-21y = 84\newliney=−4y = -4Now that we have y, we can substitute it back into the equation 4x−9y=414x - 9y = 41 to find x.\newline4x−9(−4)=414x - 9(-4) = 41\newline4x+36=414x + 36 = 41\newline−29y+z=121-29y + z = 12100\newline−29y+z=121-29y + z = 12111\newline−29y+z=121-29y + z = 12122Finally, we can substitute y=−4y = -4 into the equation −8y+z=37-8y + z = 37 to find z.\newline−29y+z=121-29y + z = 12155\newline−29y+z=121-29y + z = 12166\newline−29y+z=121-29y + z = 12177\newline−29y+z=121-29y + z = 12188

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 5 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 5 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 5 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 5 months ago

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

Posted 9 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 9 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 5 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 5 months 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 5 months 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 5 months ago

View all (22)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant