Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To solve the given system of linear equations, we can use the method of elimination to find the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex].
The given system of equations is:
[tex]\[ \begin{cases} x - 5y + 4z = 27 \\ 4x - 3y - z = 23 \\ 3x + 3y - 6z = -9 \end{cases} \][/tex]
Step 1: Eliminate [tex]\(x\)[/tex] from equations 2 and 3
We start by eliminating [tex]\(x\)[/tex] from equations (2) and (3) using equation (1).
Step 1a: Multiply equation (1) to match the coefficients of [tex]\(x\)[/tex]:
Equation (1) can be multiplied by 4 to help eliminate [tex]\(x\)[/tex] from equation (2):
[tex]\[ 4(x - 5y + 4z) = 4(27) \][/tex]
[tex]\[ 4x - 20y + 16z = 108 \][/tex] (Equation 4)
Subtract equation (4) from equation (2):
[tex]\[ (4x - 3y - z) - (4x - 20y + 16z) = 23 - 108 \][/tex]
[tex]\[ 4x - 3y - z - 4x + 20y - 16z = -85 \][/tex]
[tex]\[ 17y - 17z = -85 \][/tex]
[tex]\[ y - z = -5 \][/tex] (Equation 5)
Step 1b: Multiply equation (1) to match the coefficients of [tex]\(x\)[/tex]:
Equation (1) can be multiplied by 3 to help eliminate [tex]\(x\)[/tex] from equation (3):
[tex]\[ 3(x - 5y + 4z) = 3(27) \][/tex]
[tex]\[ 3x - 15y + 12z = 81 \][/tex] (Equation 6)
Subtract equation (6) from equation (3):
[tex]\[ (3x + 3y - 6z) - (3x - 15y + 12z) = -9 - 81 \][/tex]
[tex]\[ 3x + 3y - 6z - 3x + 15y - 12z = -90 \][/tex]
[tex]\[ 18y - 18z = -90 \][/tex]
[tex]\[ y - z = -5 \][/tex] (Equation 7)
Step 2: Solve for [tex]\(y\)[/tex] and [tex]\(z\)[/tex]
The resultant equation from Step 1 is [tex]\(y - z = -5\)[/tex]. Both are equivalent, stating:
[tex]\[ y - z = -5 \][/tex]
[tex]\[ y = z - 5 \][/tex] (Equation 8)
Step 3: Substitute [tex]\(y = z - 5\)[/tex] back into one of the original equations
Substitute [tex]\(y = z - 5\)[/tex] into Equation (1):
[tex]\[ x - 5(z - 5) + 4z = 27 \][/tex]
[tex]\[ x - 5z + 25 + 4z = 27 \][/tex]
[tex]\[ x - z + 25 = 27 \][/tex]
[tex]\[ x - z = 2 \][/tex]
[tex]\[ x = z + 2 \][/tex] (Equation 9)
Step 4: Substitute [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in another original equation to solve for [tex]\(z\)[/tex]
Substitute [tex]\(x = z + 2\)[/tex] and [tex]\(y = z - 5\)[/tex] into Equation (2):
[tex]\[ 4(z + 2) - 3(z - 5) - z = 23 \][/tex]
[tex]\[ 4z + 8 - 3z + 15 - z = 23 \][/tex]
[tex]\[ 4z - 3z - z + 8 + 15 = 23 \][/tex]
[tex]\[ 0 = 0 \][/tex]
Since this equation is always true, we have infinitely many solutions.
The general solution for the system in terms of [tex]\(z\)[/tex] is:
[tex]\[ \begin{cases} x = z + 2 \\ y = z - 5 \\ z = z \end{cases} \][/tex]
Therefore, the solution set for the system of equations [tex]\( x - 5y + 4z = 27\)[/tex], [tex]\(4x - 3y - z = 23\)[/tex], and [tex]\(3x + 3y - 6z = -9 \)[/tex] is:
[tex]\[ (x, y, z) = (z + 2, z - 5, z) \][/tex]
where [tex]\(z\)[/tex] can be any real number.
The given system of equations is:
[tex]\[ \begin{cases} x - 5y + 4z = 27 \\ 4x - 3y - z = 23 \\ 3x + 3y - 6z = -9 \end{cases} \][/tex]
Step 1: Eliminate [tex]\(x\)[/tex] from equations 2 and 3
We start by eliminating [tex]\(x\)[/tex] from equations (2) and (3) using equation (1).
Step 1a: Multiply equation (1) to match the coefficients of [tex]\(x\)[/tex]:
Equation (1) can be multiplied by 4 to help eliminate [tex]\(x\)[/tex] from equation (2):
[tex]\[ 4(x - 5y + 4z) = 4(27) \][/tex]
[tex]\[ 4x - 20y + 16z = 108 \][/tex] (Equation 4)
Subtract equation (4) from equation (2):
[tex]\[ (4x - 3y - z) - (4x - 20y + 16z) = 23 - 108 \][/tex]
[tex]\[ 4x - 3y - z - 4x + 20y - 16z = -85 \][/tex]
[tex]\[ 17y - 17z = -85 \][/tex]
[tex]\[ y - z = -5 \][/tex] (Equation 5)
Step 1b: Multiply equation (1) to match the coefficients of [tex]\(x\)[/tex]:
Equation (1) can be multiplied by 3 to help eliminate [tex]\(x\)[/tex] from equation (3):
[tex]\[ 3(x - 5y + 4z) = 3(27) \][/tex]
[tex]\[ 3x - 15y + 12z = 81 \][/tex] (Equation 6)
Subtract equation (6) from equation (3):
[tex]\[ (3x + 3y - 6z) - (3x - 15y + 12z) = -9 - 81 \][/tex]
[tex]\[ 3x + 3y - 6z - 3x + 15y - 12z = -90 \][/tex]
[tex]\[ 18y - 18z = -90 \][/tex]
[tex]\[ y - z = -5 \][/tex] (Equation 7)
Step 2: Solve for [tex]\(y\)[/tex] and [tex]\(z\)[/tex]
The resultant equation from Step 1 is [tex]\(y - z = -5\)[/tex]. Both are equivalent, stating:
[tex]\[ y - z = -5 \][/tex]
[tex]\[ y = z - 5 \][/tex] (Equation 8)
Step 3: Substitute [tex]\(y = z - 5\)[/tex] back into one of the original equations
Substitute [tex]\(y = z - 5\)[/tex] into Equation (1):
[tex]\[ x - 5(z - 5) + 4z = 27 \][/tex]
[tex]\[ x - 5z + 25 + 4z = 27 \][/tex]
[tex]\[ x - z + 25 = 27 \][/tex]
[tex]\[ x - z = 2 \][/tex]
[tex]\[ x = z + 2 \][/tex] (Equation 9)
Step 4: Substitute [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in another original equation to solve for [tex]\(z\)[/tex]
Substitute [tex]\(x = z + 2\)[/tex] and [tex]\(y = z - 5\)[/tex] into Equation (2):
[tex]\[ 4(z + 2) - 3(z - 5) - z = 23 \][/tex]
[tex]\[ 4z + 8 - 3z + 15 - z = 23 \][/tex]
[tex]\[ 4z - 3z - z + 8 + 15 = 23 \][/tex]
[tex]\[ 0 = 0 \][/tex]
Since this equation is always true, we have infinitely many solutions.
The general solution for the system in terms of [tex]\(z\)[/tex] is:
[tex]\[ \begin{cases} x = z + 2 \\ y = z - 5 \\ z = z \end{cases} \][/tex]
Therefore, the solution set for the system of equations [tex]\( x - 5y + 4z = 27\)[/tex], [tex]\(4x - 3y - z = 23\)[/tex], and [tex]\(3x + 3y - 6z = -9 \)[/tex] is:
[tex]\[ (x, y, z) = (z + 2, z - 5, z) \][/tex]
where [tex]\(z\)[/tex] can be any real number.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
