Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our 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.
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
