To determine whether the given system of equations is consistent or inconsistent, and whether it is dependent or independent, we need to analyze the system:
[tex]\[
\begin{array}{l}
y = 2x + 3 \\
x + 2y = 1
\end{array}
\][/tex]
### Step 1: Identify the Equations
We have two equations in the system:
1. \( y = 2x + 3 \)
2. \( x + 2y = 1 \)
### Step 2: Substitute \( y = 2x + 3 \) into the Second Equation
To find a solution, we substitute the expression for \( y \) from the first equation into the second equation:
[tex]\[
x + 2(2x + 3) = 1
\][/tex]
### Step 3: Simplify the Substitution
Expand and simplify the equation:
[tex]\[
x + 4x + 6 = 1
\][/tex]
Combine like terms:
[tex]\[
5x + 6 = 1
\][/tex]
Subtract 6 from both sides:
[tex]\[
5x = -5
\][/tex]
Divide by 5:
[tex]\[
x = -1
\][/tex]
### Step 4: Solve for \( y \) Using \( x = -1 \)
Substitute \( x = -1 \) back into the first equation to find \( y \):
[tex]\[
y = 2(-1) + 3
\][/tex]
Simplify:
[tex]\[
y = -2 + 3
\][/tex]
[tex]\[
y = 1
\][/tex]
### Step 5: Verify the Solution
We have found a solution \( (x, y) = (-1, 1) \). Let's verify if this solution satisfies both original equations:
Substitute \( x = -1 \) and \( y = 1 \) into the first equation:
[tex]\[
y = 2x + 3
\][/tex]
[tex]\[
1 = 2(-1) + 3
\][/tex]
[tex]\[
1 = -2 + 3
\][/tex]
[tex]\[
1 = 1 \quad (\text{True})
\][/tex]
Substitute \( x = -1 \) and \( y = 1 \) into the second equation:
[tex]\[
x + 2y = 1
\][/tex]
[tex]\[
-1 + 2(1) = 1
\][/tex]
[tex]\[
-1 + 2 = 1
\][/tex]
[tex]\[
1 = 1 \quad (\text{True})
\][/tex]
Since the solution satisfies both equations, the system is consistent.
### Step 6: Determine Independence or Dependence
To determine if the system is independent or dependent, we need to check if there is only one unique solution:
The system has a unique solution \( (x, y) = (-1, 1) \).
Therefore, the system is independent.
### Conclusion
The system of equations is consistent and independent. The unique solution to the system is:
[tex]\[
x = -1, \quad y = 1
\][/tex]
