To solve the given system of equations by substitution, the process involves isolating one variable in one of the equations and substituting it into the other equation. Here's a detailed, step-by-step solution to determine the best variable and equation for substitution:
Given the system of equations:
1. [tex]\( x + 6y = 9 \)[/tex]
2. [tex]\( x - 10y = 13 \)[/tex]
### Step-by-Step Solution:
1. Consider isolating one variable:
- First equation: [tex]\( x + 6y = 9 \)[/tex]
- Second equation: [tex]\( x - 10y = 13 \)[/tex]
2. Solve for [tex]\( x \)[/tex] in the first equation:
- [tex]\( x + 6y = 9 \)[/tex]
- To isolate [tex]\( x \)[/tex], subtract [tex]\( 6y \)[/tex] from both sides:
[tex]\[
x = 9 - 6y
\][/tex]
3. Substitute [tex]\( x \)[/tex] from the first equation into the second equation:
- Second equation: [tex]\( x - 10y = 13 \)[/tex]
- Substitute [tex]\( x = 9 - 6y \)[/tex]:
[tex]\[
(9 - 6y) - 10y = 13
\][/tex]
4. Simplify and solve for [tex]\( y \)[/tex]:
- Combine like terms:
[tex]\[
9 - 16y = 13
\][/tex]
- Subtract 9 from both sides:
[tex]\[
-16y = 4
\][/tex]
- Divide both sides by -16:
[tex]\[
y = -\frac{1}{4}
\][/tex]
### Conclusion:
From the detailed steps, the best way to solve this system is to isolate [tex]\( x \)[/tex] in the first equation [tex]\( x + 6y = 9 \)[/tex] and then substitute it into the second equation [tex]\( x - 10y = 13 \)[/tex].
Thus, the best variable to solve for is [tex]\( \mathbf{x} \)[/tex], and the best equation to solve from is the [tex]\(\mathbf{first \ equation}\)[/tex].
Answer: [tex]\( \boxed{\text{B. } x, \text{ in the first equation}} \)[/tex]
