Answer :
Let's classify each of the given systems. We have three different systems of linear equations, and we need to determine whether each system is consistent, inconsistent, or dependent.
### 1. First System:
[tex]\[ \begin{cases} x + 3y = 4 \\ 3x + 9y = 12 \end{cases} \][/tex]
To classify this system, let's analyze the equations:
#### Step 1: Check for Proportionality
First, notice that the second equation is a multiple of the first equation. If we multiply the entire first equation by 3, we get:
[tex]\[ 3(x + 3y) = 3 \cdot 4 \implies 3x + 9y = 12 \][/tex]
This shows that the second equation is just the first equation scaled by a factor of 3. Therefore, both equations represent the same line.
#### Conclusion:
Since both equations represent the same line, the system has infinitely many solutions (all points on the line). This system is dependent.
### 2. Second System:
[tex]\[ \begin{cases} 3x - 4y = 12 \\ 6x - 8y = 21 \end{cases} \][/tex]
Let's analyze these equations:
#### Step 1: Check for Proportionality
Multiply the first equation by 2:
[tex]\[ 2(3x - 4y) = 2 \cdot 12 \implies 6x - 8y = 24 \][/tex]
Notice that [tex]\(6x - 8y = 24\)[/tex] is not equal to the second equation [tex]\(6x - 8y = 21\)[/tex].
#### Step 2: Compare the Resulting Equations
We see:
- [tex]\(6x - 8y = 24\)[/tex]
- [tex]\(6x - 8y = 21\)[/tex]
The resulting equations have the same coefficients, i.e., they are proportional, but they have different constants resulting in a contradiction. These lines are parallel and do not intersect.
#### Conclusion:
The system has no solution because it represents parallel lines that do not intersect. This system is inconsistent.
### 3. Third System:
[tex]\[ \begin{cases} 2x - 3y = 8 \\ -3x + 2y = 8 \end{cases} \][/tex]
#### Step 1: Solve Linear System
To solve the system, let's use the method of elimination or substitution. First, let's multiply each equation by suitable numbers so that the coefficients of [tex]\(x\)[/tex] or [tex]\(y\)[/tex] will cancel out when added.
Multiply the first equation by 3 and the second equation by 2:
[tex]\[ 3(2x - 3y) = 3 \cdot 8 \implies 6x - 9y = 24 \][/tex]
[tex]\[ 2(-3x + 2y) = 2 \cdot 8 \implies -6x + 4y = 16 \][/tex]
Now add these two equations to eliminate [tex]\(x\)[/tex]:
[tex]\[ 6x - 9y - 6x + 4y = 24 + 16 \][/tex]
[tex]\[ -5y = 40 \implies y = -8 \][/tex]
Now substitute [tex]\(y = -8\)[/tex] back into the first equation to solve for [tex]\(x\)[/tex]:
[tex]\[ 2x - 3(-8) = 8 \implies 2x + 24 = 8 \implies 2x = -16 \implies x = -8 \][/tex]
So, we have [tex]\(x = -8\)[/tex] and [tex]\(y = -8\)[/tex].
#### Conclusion:
The system has a unique solution [tex]\((x, y) = (-8, -8)\)[/tex]. This system is consistent.
### Summary:
- First System: Dependent (infinitely many solutions)
- Second System: Inconsistent (no solution)
- Third System: Consistent (unique solution)
### 1. First System:
[tex]\[ \begin{cases} x + 3y = 4 \\ 3x + 9y = 12 \end{cases} \][/tex]
To classify this system, let's analyze the equations:
#### Step 1: Check for Proportionality
First, notice that the second equation is a multiple of the first equation. If we multiply the entire first equation by 3, we get:
[tex]\[ 3(x + 3y) = 3 \cdot 4 \implies 3x + 9y = 12 \][/tex]
This shows that the second equation is just the first equation scaled by a factor of 3. Therefore, both equations represent the same line.
#### Conclusion:
Since both equations represent the same line, the system has infinitely many solutions (all points on the line). This system is dependent.
### 2. Second System:
[tex]\[ \begin{cases} 3x - 4y = 12 \\ 6x - 8y = 21 \end{cases} \][/tex]
Let's analyze these equations:
#### Step 1: Check for Proportionality
Multiply the first equation by 2:
[tex]\[ 2(3x - 4y) = 2 \cdot 12 \implies 6x - 8y = 24 \][/tex]
Notice that [tex]\(6x - 8y = 24\)[/tex] is not equal to the second equation [tex]\(6x - 8y = 21\)[/tex].
#### Step 2: Compare the Resulting Equations
We see:
- [tex]\(6x - 8y = 24\)[/tex]
- [tex]\(6x - 8y = 21\)[/tex]
The resulting equations have the same coefficients, i.e., they are proportional, but they have different constants resulting in a contradiction. These lines are parallel and do not intersect.
#### Conclusion:
The system has no solution because it represents parallel lines that do not intersect. This system is inconsistent.
### 3. Third System:
[tex]\[ \begin{cases} 2x - 3y = 8 \\ -3x + 2y = 8 \end{cases} \][/tex]
#### Step 1: Solve Linear System
To solve the system, let's use the method of elimination or substitution. First, let's multiply each equation by suitable numbers so that the coefficients of [tex]\(x\)[/tex] or [tex]\(y\)[/tex] will cancel out when added.
Multiply the first equation by 3 and the second equation by 2:
[tex]\[ 3(2x - 3y) = 3 \cdot 8 \implies 6x - 9y = 24 \][/tex]
[tex]\[ 2(-3x + 2y) = 2 \cdot 8 \implies -6x + 4y = 16 \][/tex]
Now add these two equations to eliminate [tex]\(x\)[/tex]:
[tex]\[ 6x - 9y - 6x + 4y = 24 + 16 \][/tex]
[tex]\[ -5y = 40 \implies y = -8 \][/tex]
Now substitute [tex]\(y = -8\)[/tex] back into the first equation to solve for [tex]\(x\)[/tex]:
[tex]\[ 2x - 3(-8) = 8 \implies 2x + 24 = 8 \implies 2x = -16 \implies x = -8 \][/tex]
So, we have [tex]\(x = -8\)[/tex] and [tex]\(y = -8\)[/tex].
#### Conclusion:
The system has a unique solution [tex]\((x, y) = (-8, -8)\)[/tex]. This system is consistent.
### Summary:
- First System: Dependent (infinitely many solutions)
- Second System: Inconsistent (no solution)
- Third System: Consistent (unique solution)
