How many solutions does the system of equations below have? $\newline$ $-2x - 2y - z = 3$ $\newline$ $x - 3y + z = 17$ $\newline$ $3x + 2y - z = -13$ $\newline$ Choices: $\newline$ (A)no solution $\newline$ (B)one solution $\newline$ (C)infinitely many solutions

View step-by-step help

Home

Math Problems

Algebra 2

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

Full solution

Q. How many solutions does the system of equations below have?

\newline

-2x - 2y - z = 3

\newline

x - 3y + z = 17

\newline

3x + 2y - z = -13

\newline

Choices:

\newline

(A)no solution

\newline

(B)one solution

\newline

(C)infinitely many solutions

Eliminate z: First, let's add the first and second equations to eliminate z. $\newline$ $(-2x - 2y - z) + (x - 3y + z) = 3 + 17$ $\newline$ $-2x - 2y - z + x - 3y + z = 20$ $\newline$ $-2x + x - 2y - 3y = 20$ $\newline$ $-x - 5y = 20$
Eliminate z again: Now, let's multiply the second equation by $2$ and add it to the third equation to also eliminate z. $\newline$ $2(x - 3y + z) + (3x + 2y - z) = 2(17) + (-13)$ $\newline$ $2x - 6y + 2z + 3x + 2y - z = 34 - 13$ $\newline$ $2x + 3x - 6y + 2y + 2z - z = 21$ $\newline$ $5x - 4y + z = 21$
Eliminate y: We can now add the modified first equation to the modified third equation to eliminate y. $\newline$ $(-x - 5y) + (5x - 4y) = 20 + 21$ $\newline$ $-x + 5x - 5y - 4y = 41$ $\newline$ $4x - 9y = 41$
Solve for x: Let's solve for x using the equation $4x - 9y = 41$ . Since we don't have a value for $y$ , we can't solve for $x$ directly. We need to find another equation that relates $x$ and $y$ to solve the system.
Find $x$ and $y$ : We can use the modified first equation $-x - 5y = 20$ and the modified third equation $5x - 4y + z = 21$ to solve for $x$ and $y$ . Let's multiply the first equation by $5$ to match the coefficient of $x$ in the third equation. $5(-x - 5y) = 5(20)$ $-5x - 25y = 100$
Eliminate x: Now, let's add this new equation to the third equation to eliminate x. $\newline$ $(-5x - 25y) + (5x - 4y + z) = 100 + 21$ $\newline$ $-5x + 5x - 25y - 4y + z = 121$ $\newline$ $-29y + z = 121$
Solve for $y$ : We now have two equations with two variables: $\newline$ $4x - 9y = 41$ $\newline$ $-29y + z = 121$ $\newline$ We can't solve for a specific value of $y$ or $z$ without another equation relating them. We need to go back and check our steps to see if we can find another relationship.
Find x: Looking back at the original equations, we can multiply the second equation by $2$ and add it to the first to eliminate z. $\newline$ $2(x - 3y + z) + (-2x - 2y - z) = 2(17) + 3$ $\newline$ $2x - 6y + 2z - 2x - 2y - z = 34 + 3$ $\newline$ $-6y - 2y + 2z - z = 37$ $\newline$ $-8y + z = 37$
Find $z$ : We now have three equations with two variables: $\newline$ $4x - 9y = 41$ $\newline$ $-29y + z = 121$ $\newline$ $-8y + z = 37$ $\newline$ We can use the last two equations to solve for $y$ and $z$ .
Find $z$ : We now have three equations with two variables: $\newline$ $4x - 9y = 41$ $\newline$ $-29y + z = 121$ $\newline$ $-8y + z = 37$ $\newline$ We can use the last two equations to solve for $y$ and $z$ .Subtract the third equation from the second to eliminate $z$ . $\newline$ $(-29y + z) - (-8y + z) = 121 - 37$ $\newline$ $-29y + z + 8y - z = 84$ $\newline$ $-21y = 84$ $\newline$ $y = -4$
Find $z$ : We now have three equations with two variables: $\newline$ $4x - 9y = 41$ $\newline$ $-29y + z = 121$ $\newline$ $-8y + z = 37$ $\newline$ We can use the last two equations to solve for $y$ and $z$ .Subtract the third equation from the second to eliminate $z$ . $\newline$ $(-29y + z) - (-8y + z) = 121 - 37$ $\newline$ $-29y + z + 8y - z = 84$ $\newline$ $-21y = 84$ $\newline$ $y = -4$ Now that we have $y$ , we can substitute it back into the equation $4x - 9y = 41$ to find $x$ . $\newline$ $4x - 9(-4) = 41$ $\newline$ $4x + 36 = 41$ $\newline$ $4x = 41 - 36$ $\newline$ $-29y + z = 121$ $0$ $\newline$ $-29y + z = 121$ $1$
Find z: We now have three equations with two variables: $\newline$ $4x - 9y = 41$ $\newline$ $-29y + z = 121$ $\newline$ $-8y + z = 37$ $\newline$ We can use the last two equations to solve for y and z.Subtract the third equation from the second to eliminate z. $\newline$ $(-29y + z) - (-8y + z) = 121 - 37$ $\newline$ $-29y + z + 8y - z = 84$ $\newline$ $-21y = 84$ $\newline$ $y = -4$ Now that we have y, we can substitute it back into the equation $4x - 9y = 41$ to find x. $\newline$ $4x - 9(-4) = 41$ $\newline$ $4x + 36 = 41$ $\newline$ $-29y + z = 121$ $0$ $\newline$ $-29y + z = 121$ $1$ $\newline$ $-29y + z = 121$ $2$ Finally, we can substitute $y = -4$ into the equation $-8y + z = 37$ to find z. $\newline$ $-29y + z = 121$ $5$ $\newline$ $-29y + z = 121$ $6$ $\newline$ $-29y + z = 121$ $7$ $\newline$ $-29y + z = 121$ $8$