6-64. Kevin and his little sister, Katy, are trying to solve the system of equations shown below. Kevin thinks that using the Substitution Method should give a new equation of 3 \( (6 x-1)+2 y=43 \), while Katy thinks it should be \( 3 x+2(6 x-1)=43 \). Who is correct and why? \[ y=6 x-1 \] \( 3 x+2 y=43 \)

Katy is correct in her approach to the Substitution Method. Here's why: **Given System of Equations:** 1. \( y = 6x - 1 \)  — *Equation 1* 2. \( 3x + 2y = 43 \)   — *Equation 2* **Substitution Method Steps:** 1. **Solve one equation for one variable.** Equation 1 already expresses \( y \) in terms of \( x \): \[ y = 6x - 1 \] 2. **Substitute this expression into the other equation.** Substitute \( y = 6x - 1 \) into Equation 2: \[ 3x + 2(6x - 1) = 43 \] This is precisely what Katy did. **Why Kevin's Approach is Incorrect:** Kevin wrote: \[ 3(6x - 1) + 2y = 43 \] This introduces a substitution loop by substituting \( y \) back into Equation 2 **without eliminating** one variable. Instead of reducing the system to a single equation with one variable (which is the goal of the Substitution Method), Kevin's equation still contains both \( x \) and \( y \), making it more complicated to solve. **Conclusion:** Katy correctly substituted \( y = 6x - 1 \) into the second equation, simplifying the system to a single equation with one variable, thereby effectively applying the Substitution Method. **Answer:** Katy is correct because she correctly substituted y = 6x – 1 into the second equation, obtaining 3x + 2(6x − 1) = 43.