At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To solve the given system of equations, we have:
[tex]\[ \begin{aligned} 1: & \quad 2x + 4y - 3z = 12 \\ 2: & \quad -3x - 2y + 2z = -9 \\ 3: & \quad -x + y - 3z = 0 \end{aligned} \][/tex]
We need to find values for [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations. Let's proceed step-by-step.
### Step 1: Solve for One Variable
Begin with equation [tex]\(3\)[/tex]:
[tex]\[ -x + y - 3z = 0 \][/tex]
Rearrange to solve for one variable, say [tex]\( y \)[/tex]:
[tex]\[ y = x + 3z \quad \text{(Equation 3')} \][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] into Equations 1 and 2
Substitute [tex]\( y = x + 3z \)[/tex] into equations 1 and 2:
#### For Equation 1:
[tex]\[ \begin{aligned} 2x + 4y - 3z &= 12 \\ 2x + 4(x + 3z) - 3z &= 12 \\ 2x + 4x + 12z - 3z &= 12 \\ 6x + 9z &= 12 \\ \end{aligned} \][/tex]
Simplify:
[tex]\[ 6x + 9z = 12 \quad \text{(Equation 1')} \][/tex]
Divide the entire equation by 3:
[tex]\[ 2x + 3z = 4 \quad \text{(Equation 1'')} \][/tex]
#### For Equation 2:
[tex]\[ \begin{aligned} -3x - 2y + 2z &= -9 \\ -3x - 2(x + 3z) + 2z &= -9 \\ -3x - 2x - 6z + 2z &= -9 \\ -5x - 4z &= -9 \\ \end{aligned} \][/tex]
Simplify:
[tex]\[ -5x - 4z = -9 \quad \text{(Equation 2')} \][/tex]
### Step 3: Solve the Simplified System of Two Variables
Now we have a simpler system of two equations with two variables:
[tex]\[ \begin{aligned} 1'': & \quad 2x + 3z = 4 \\ 2': & \quad -5x - 4z = -9 \end{aligned} \][/tex]
Let's solve these equations simultaneously.
#### Solve Equation 1'' for [tex]\( x \)[/tex]:
[tex]\[ 2x + 3z = 4 \\ 2x = 4 - 3z \\ x = \frac{4 - 3z}{2} \quad \text{(Equation 4)} \][/tex]
#### Substitute Equation 4 into Equation 2':
[tex]\[ -5 \left( \frac{4 - 3z}{2} \right) - 4z = -9 \\ - \frac{20 - 15z}{2} - 4z = -9 \\ -10 + \frac{15z}{2} - 4z = -9 \\ -10 + \frac{15z}{2} - \frac{8z}{2} = -9 \\ -10 + \frac{7z}{2} = -9 \\ \frac{7z}{2} = 1 \\ 7z = 2 \\ z = \frac{2}{7} \][/tex]
### Step 4: Substitute [tex]\( z \)[/tex] into Equation 4 to Find [tex]\( x \)[/tex]:
[tex]\[ x = \frac{4 - 3z}{2} \\ x = \frac{4 - 3 \left( \frac{2}{7} \right)}{2} \\ x = \frac{4 - \frac{6}{7}}{2} \\ x = \frac{\frac{28}{7} - \frac{6}{7}}{2} \\ x = \frac{\frac{22}{7}}{2} \\ x = \frac{22}{14} \\ x = \frac{11}{7} \][/tex]
### Step 5: Substitute [tex]\( x \)[/tex] and [tex]\( z \)[/tex] into Equation 3' to Find [tex]\( y \)[/tex]:
[tex]\[ y = x + 3z \\ y = \frac{11}{7} + 3 \left( \frac{2}{7} \right) \\ y = \frac{11}{7} + \frac{6}{7} \\ y = \frac{17}{7} \][/tex]
### Final Solution:
The solutions to the system of equations are:
[tex]\[ x = \frac{11}{7}, \quad y = \frac{17}{7}, \quad z = \frac{2}{7} \][/tex]
These values satisfy all the given equations.
[tex]\[ \begin{aligned} 1: & \quad 2x + 4y - 3z = 12 \\ 2: & \quad -3x - 2y + 2z = -9 \\ 3: & \quad -x + y - 3z = 0 \end{aligned} \][/tex]
We need to find values for [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations. Let's proceed step-by-step.
### Step 1: Solve for One Variable
Begin with equation [tex]\(3\)[/tex]:
[tex]\[ -x + y - 3z = 0 \][/tex]
Rearrange to solve for one variable, say [tex]\( y \)[/tex]:
[tex]\[ y = x + 3z \quad \text{(Equation 3')} \][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] into Equations 1 and 2
Substitute [tex]\( y = x + 3z \)[/tex] into equations 1 and 2:
#### For Equation 1:
[tex]\[ \begin{aligned} 2x + 4y - 3z &= 12 \\ 2x + 4(x + 3z) - 3z &= 12 \\ 2x + 4x + 12z - 3z &= 12 \\ 6x + 9z &= 12 \\ \end{aligned} \][/tex]
Simplify:
[tex]\[ 6x + 9z = 12 \quad \text{(Equation 1')} \][/tex]
Divide the entire equation by 3:
[tex]\[ 2x + 3z = 4 \quad \text{(Equation 1'')} \][/tex]
#### For Equation 2:
[tex]\[ \begin{aligned} -3x - 2y + 2z &= -9 \\ -3x - 2(x + 3z) + 2z &= -9 \\ -3x - 2x - 6z + 2z &= -9 \\ -5x - 4z &= -9 \\ \end{aligned} \][/tex]
Simplify:
[tex]\[ -5x - 4z = -9 \quad \text{(Equation 2')} \][/tex]
### Step 3: Solve the Simplified System of Two Variables
Now we have a simpler system of two equations with two variables:
[tex]\[ \begin{aligned} 1'': & \quad 2x + 3z = 4 \\ 2': & \quad -5x - 4z = -9 \end{aligned} \][/tex]
Let's solve these equations simultaneously.
#### Solve Equation 1'' for [tex]\( x \)[/tex]:
[tex]\[ 2x + 3z = 4 \\ 2x = 4 - 3z \\ x = \frac{4 - 3z}{2} \quad \text{(Equation 4)} \][/tex]
#### Substitute Equation 4 into Equation 2':
[tex]\[ -5 \left( \frac{4 - 3z}{2} \right) - 4z = -9 \\ - \frac{20 - 15z}{2} - 4z = -9 \\ -10 + \frac{15z}{2} - 4z = -9 \\ -10 + \frac{15z}{2} - \frac{8z}{2} = -9 \\ -10 + \frac{7z}{2} = -9 \\ \frac{7z}{2} = 1 \\ 7z = 2 \\ z = \frac{2}{7} \][/tex]
### Step 4: Substitute [tex]\( z \)[/tex] into Equation 4 to Find [tex]\( x \)[/tex]:
[tex]\[ x = \frac{4 - 3z}{2} \\ x = \frac{4 - 3 \left( \frac{2}{7} \right)}{2} \\ x = \frac{4 - \frac{6}{7}}{2} \\ x = \frac{\frac{28}{7} - \frac{6}{7}}{2} \\ x = \frac{\frac{22}{7}}{2} \\ x = \frac{22}{14} \\ x = \frac{11}{7} \][/tex]
### Step 5: Substitute [tex]\( x \)[/tex] and [tex]\( z \)[/tex] into Equation 3' to Find [tex]\( y \)[/tex]:
[tex]\[ y = x + 3z \\ y = \frac{11}{7} + 3 \left( \frac{2}{7} \right) \\ y = \frac{11}{7} + \frac{6}{7} \\ y = \frac{17}{7} \][/tex]
### Final Solution:
The solutions to the system of equations are:
[tex]\[ x = \frac{11}{7}, \quad y = \frac{17}{7}, \quad z = \frac{2}{7} \][/tex]
These values satisfy all the given equations.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
