Certainly! Let's analyze which equations can pair with [tex]\( y = 3x - 2 \)[/tex] to create a consistent and independent system.
1. Equation: [tex]\( x = 3y - 2 \)[/tex]
- Let's solve the system:
[tex]\[
y = 3x - 2 \quad \text{and} \quad x = 3y - 2
\][/tex]
- This system provides a solution in terms of the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex], resulting in:
[tex]\[
x = 3y - 2 \quad \text{and} \quad y = 9y - 8
\][/tex]
- Thus, the solution is consistent and independent:
[tex]\[
\{ x = 3y - 2, y = 9y - 8 \}
\][/tex]
2. Equation: [tex]\( y = -3x - 2 \)[/tex]
- Let's solve the system:
[tex]\[
y = 3x - 2 \quad \text{and} \quad y = -3x - 2
\][/tex]
- This system provides a specific numeric solution:
[tex]\[
x = 0 \quad \text{and} \quad y = -2
\][/tex]
- Thus, the solution is consistent and independent:
[tex]\[
\{ x = 0, y = -2 \}
\][/tex]
3. Equation: [tex]\( y = 3x + 2 \)[/tex]
- Let's solve the system:
[tex]\[
y = 3x - 2 \quad \text{and} \quad y = 3x + 2
\][/tex]
- These two equations contradict each other, implying no solution:
[tex]\[
\square \text{(no solution)}
\][/tex]
4. Equation: [tex]\( 6x - 2y = 4 \)[/tex]
- Let's solve the system:
[tex]\[
y = 3x - 2 \quad \text{and} \quad 6x - 2y = 4
\][/tex]
- This system can be solved to give a relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \frac{y}{3} + \frac{2}{3}
\][/tex]
- Thus, it results in a consistent and independent solution:
[tex]\[
\{ x = \frac{y}{3} + \frac{2}{3} \}
\][/tex]
5. Equation: [tex]\( 3y - x = -2 \)[/tex]
- Let's solve the system:
[tex]\[
y = 3x - 2 \quad \text{and} \quad 3y - x = -2
\][/tex]
- This system provides a specific numeric solution:
[tex]\[
x = \frac{1}{2} \quad \text{and} \quad y = -\frac{1}{2}
\][/tex]
- Thus, the solution is consistent and independent:
[tex]\[
\{ x = \frac{1}{2}, y = -\frac{1}{2} \}
\][/tex]
Based on the solutions found, the equations that can pair with [tex]\( y = 3x - 2 \)[/tex] to create a consistent and independent system are:
- [tex]\( x = 3y - 2 \)[/tex]
- [tex]\( y = -3x - 2 \)[/tex]
- [tex]\( 6x - 2y = 4 \)[/tex]
- [tex]\( 3y - x = -2 \)[/tex]
The pair [tex]\( y = 3x + 2 \)[/tex] does not create a consistent and independent system when paired with [tex]\( y = 3x - 2 \)[/tex].
