How many solutions does the system of equations below have? x + 3y + 2z = 5 x + 2y - 2z = 16 -2x + 2y - z = -11 Choices: [[no solution][one solution][infinitely many solutions]]

Question

How many solutions does the system of equations below have?  x + 3y + 2z = 5  x + 2y - 2z = 16  -2x + 2y - z = -11            Choices: [[no solution][one solution][infinitely many solutions]]

Bytelearn · Accepted Answer

Write Equations: Write down the system of equations to analyze the structure. x + 3y + 2z = 5 x + 2y - 2z = 16 -2x + 2y - z = -11Eliminate x: Subtract the second equation from the first to eliminate x and find an equation in terms of y and z. (1) - (2): (x + 3y + 2z) - (x + 2y - 2z) = 5 - 16 This simplifies to y + 4z = -11Simplify Coefficients: Multiply the third equation by 0.5 to simplify the coefficients. 0.5 * (-2x + 2y - z) = 0.5 * (-11) This simplifies to -x + y - 0.5z = -5.5Eliminate x Again: Add the simplified third equation to the second equation to eliminate x and find another equation in terms of y and z. (2) + (3): (x + 2y - 2z) + (-x + y - 0.5z) = 16 - 5.5 This simplifies to 3y - 2.5z = 10.5Solve for y and z: We now have two equations with two variables: y + 4z = -11 3y - 2.5z = 10.5 We can use these two equations to solve for y and z.Align y Terms: Multiply the first of these two equations by 3 to align the y terms: 3(y + 4z) = 3(-11) This simplifies to 3y + 12z = -33Eliminate y: Subtract the new equation from the second equation to eliminate y: (3y + 12z) - (3y - 2.5z) = -33 - 10.5 This simplifies to 14.5z = -43.5Solve for z: Solve for z: 14.5z = -43.5 z = -43.5 / 14.5 z = -3Solve for y: Substitute z = -3 into one of the two-variable equations to solve for y: y + 4(-3) = -11 y - 12 = -11 y = -11 + 12 y = 1Substitute for x: Substitute y = 1 and z = -3 into one of the original equations to solve for x: x + 3(1) + 2(-3) = 5 x + 3 - 6 = 5 x - 3 = 5 x = 5 + 3 x = 8Check Solution: Check the solution (x = 8, y = 1, z = -3) in all three original equations to ensure it satisfies all of them. First equation: 8 + 3(1) + 2(-3) = 5 Second equation: 8 + 2(1) - 2(-3) = 16 Third equation: -2(8) + 2(1) - (-3) = -11Verify the solutions in the original equations: First equation: 8 + 3 + (-6) = 5, which is true. Second equation: 8 + 2 + 6 = 16, which is true. Third equation: -16 + 2 + 3 = -11, which is true. Since the solution satisfies all three equations, the system has one solution.

How many solutions does the system of equations below have? $\newline$ $x + 3y + 2z = 5$ $\newline$ $x + 2y - 2z = 16$ $\newline$ $-2x + 2y - z = -11$ $\newline$ Choices: $\newline$ $\text{[A]no solution}$ $\newline$ $\text{[B]one solution}$ $\newline$ $\text{[C]infinitely many solutions}$

Full solution

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

Related topics