Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Experience the convenience of getting reliable answers to your questions from a vast network of knowledgeable experts. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To solve the system of linear equations given in terms of the nonzero constants [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \begin{aligned} &1) \quad a x - b y - 2 c z = 15, \\ &2) \quad a x + b y + c z = 0, \\ &3) \quad 7 a x - b y + c z = 6, \end{aligned} \][/tex]
we will find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex].
Step-by-Step Solution:
### Equation Setup
First, we label the equations for reference:
[tex]\[ \begin{aligned} & (1) \quad a x - b y - 2 c z = 15, \\ & (2) \quad a x + b y + c z = 0, \\ & (3) \quad 7 a x - b y + c z = 6. \end{aligned} \][/tex]
### Step 1: Combine Equations
Combine equations to eliminate variables and isolate one.
Combine (1) and (2):
Adding (1) and (2):
[tex]\[ (a x - b y - 2 c z) + (a x + b y + c z) = 15 + 0, \][/tex]
[tex]\[ 2 a x - c z = 15. \][/tex]
This gives us:
[tex]\[ 2 a x - c z = 15 \quad \text{(4)}. \][/tex]
Combine (2) and (3):
Adding (2) and (3):
[tex]\[ (a x + b y + c z) + (7 a x - b y + c z) = 0 + 6, \][/tex]
[tex]\[ 8 a x + 2 c z = 6. \][/tex]
This gives us:
[tex]\[ 4 a x + c z = 3 \quad \text{(5)}. \][/tex]
### Step 2: Solve for [tex]\( x \)[/tex] and [tex]\( z \)[/tex]
Next, we have two new equations (4) and (5):
[tex]\[ \begin{aligned} & (4) \quad 2 a x - c z = 15, \\ & (5) \quad 4 a x + c z = 3. \end{aligned} \][/tex]
Add equations (4) and (5):
[tex]\[ (2 a x - c z) + (4 a x + c z) = 15 + 3, \][/tex]
[tex]\[ 6 a x = 18, \][/tex]
[tex]\[ x = \frac{18}{6 a} = \frac{3}{a}. \][/tex]
Solve for [tex]\( z \)[/tex] using the value of [tex]\( x \)[/tex] in (5):
[tex]\[ 4 a \left(\frac{3}{a}\right) + c z = 3, \][/tex]
[tex]\[ 4 \cdot 3 + c z = 3, \][/tex]
[tex]\[ 12 + c z = 3, \][/tex]
[tex]\[ c z = 3 - 12 = -9, \][/tex]
[tex]\[ z = \frac{-9}{c}. \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
Substitute [tex]\( x = \frac{3}{a} \)[/tex] and [tex]\( z = \frac{-9}{c} \)[/tex] into equation (2):
[tex]\[ a \left(\frac{3}{a}\right) + b y + c \left(\frac{-9}{c}\right) = 0, \][/tex]
[tex]\[ 3 + b y - 9 = 0, \][/tex]
[tex]\[ b y = 6, \][/tex]
[tex]\[ y = \frac{6}{b}. \][/tex]
### Solution Set
Thus, the solution set for the system is:
[tex]\[ \left( x, y, z \right) = \left( \frac{3}{a}, \frac{6}{b}, \frac{-9}{c} \right). \][/tex]
So, the solution set is [tex]\(\left\{ \left( \frac{3}{a}, \frac{6}{b}, \frac{-9}{c} \right) \right\}\)[/tex].
[tex]\[ \begin{aligned} &1) \quad a x - b y - 2 c z = 15, \\ &2) \quad a x + b y + c z = 0, \\ &3) \quad 7 a x - b y + c z = 6, \end{aligned} \][/tex]
we will find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex].
Step-by-Step Solution:
### Equation Setup
First, we label the equations for reference:
[tex]\[ \begin{aligned} & (1) \quad a x - b y - 2 c z = 15, \\ & (2) \quad a x + b y + c z = 0, \\ & (3) \quad 7 a x - b y + c z = 6. \end{aligned} \][/tex]
### Step 1: Combine Equations
Combine equations to eliminate variables and isolate one.
Combine (1) and (2):
Adding (1) and (2):
[tex]\[ (a x - b y - 2 c z) + (a x + b y + c z) = 15 + 0, \][/tex]
[tex]\[ 2 a x - c z = 15. \][/tex]
This gives us:
[tex]\[ 2 a x - c z = 15 \quad \text{(4)}. \][/tex]
Combine (2) and (3):
Adding (2) and (3):
[tex]\[ (a x + b y + c z) + (7 a x - b y + c z) = 0 + 6, \][/tex]
[tex]\[ 8 a x + 2 c z = 6. \][/tex]
This gives us:
[tex]\[ 4 a x + c z = 3 \quad \text{(5)}. \][/tex]
### Step 2: Solve for [tex]\( x \)[/tex] and [tex]\( z \)[/tex]
Next, we have two new equations (4) and (5):
[tex]\[ \begin{aligned} & (4) \quad 2 a x - c z = 15, \\ & (5) \quad 4 a x + c z = 3. \end{aligned} \][/tex]
Add equations (4) and (5):
[tex]\[ (2 a x - c z) + (4 a x + c z) = 15 + 3, \][/tex]
[tex]\[ 6 a x = 18, \][/tex]
[tex]\[ x = \frac{18}{6 a} = \frac{3}{a}. \][/tex]
Solve for [tex]\( z \)[/tex] using the value of [tex]\( x \)[/tex] in (5):
[tex]\[ 4 a \left(\frac{3}{a}\right) + c z = 3, \][/tex]
[tex]\[ 4 \cdot 3 + c z = 3, \][/tex]
[tex]\[ 12 + c z = 3, \][/tex]
[tex]\[ c z = 3 - 12 = -9, \][/tex]
[tex]\[ z = \frac{-9}{c}. \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
Substitute [tex]\( x = \frac{3}{a} \)[/tex] and [tex]\( z = \frac{-9}{c} \)[/tex] into equation (2):
[tex]\[ a \left(\frac{3}{a}\right) + b y + c \left(\frac{-9}{c}\right) = 0, \][/tex]
[tex]\[ 3 + b y - 9 = 0, \][/tex]
[tex]\[ b y = 6, \][/tex]
[tex]\[ y = \frac{6}{b}. \][/tex]
### Solution Set
Thus, the solution set for the system is:
[tex]\[ \left( x, y, z \right) = \left( \frac{3}{a}, \frac{6}{b}, \frac{-9}{c} \right). \][/tex]
So, the solution set is [tex]\(\left\{ \left( \frac{3}{a}, \frac{6}{b}, \frac{-9}{c} \right) \right\}\)[/tex].
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.