Question

Jon has to choose which variable to solve for in order to be able to do the problem below in the most efficient manner.

6 x + 3 y = 27. 5 x + 2 y = 21.

Which variable should he choose so that he can use substitution to solve the system?
Jon should solve for y in the first equation because the coefficients can be reduced by a common factor to eliminate the coefficient for y.
Jon should solve for x in the first equation because the coefficients can be reduced by a common factor to eliminate the coefficient for x.
Jon should solve for y in the second equation because the coefficients can be reduced by a common factor to eliminate the coefficient for y.
Jon should solve for x in the second equation because the coefficients can be reduced by a common factor to eliminate the coefficient for x.

Answer 1

To determine the most efficient variable for Jon to solve for using substitution in the given system of equations:

1. **Equations:**
- \(6x + 3y = 27\) (Equation 1)
- \(5x + 2y = 21\) (Equation 2)

2. **Simplifying the Equations:**
- We can simplify Equation 1 by dividing all terms by 3:
\[
2x + y = 9 \quad \text{(Simplified Equation 1)}
\]
- Equation 2 does not have common factors that would simplify directly, as all terms do not have a common factor.

3. **Preference for Substitution:**
- From the simplified Equation 1, we can easily isolate \(y\):
\[
y = 9 - 2x
\]

4. **Determining Efficiency:**
- This form allows us to easily substitute this expression into Equation 2:
\[
5x + 2(9 - 2x) = 21
\]
- The substitution leads to a straightforward equation to solve for \(x\).

Considering the above analysis, the best choice for Jon would be:

**Jon should solve for y in the first equation because the coefficients can be reduced by a common factor to eliminate the coefficient for y.**

This allows for an efficient substitution into the second equation.