To approach solving the system of equations by substitution, we should aim to isolate one of the variables from one of the equations first. The goal is to simplify our work, so we search for an equation from which isolating a variable would require the least complexity. Let's examine both equations in detail:
Given system of equations:
[tex]\[
\begin{array}{c}
3x + 6y = 9 \\
2x - 10y = 13
\end{array}
\][/tex]
First, let’s consider isolating [tex]\( y \)[/tex] from the first equation:
[tex]\[ 3x + 6y = 9 \][/tex]
We can isolate [tex]\( y \)[/tex] as follows:
[tex]\[ 6y = 9 - 3x \][/tex]
[tex]\[ y = \frac{9 - 3x}{6} \][/tex]
[tex]\[ y = \frac{3(3 - x)}{6} \][/tex]
[tex]\[ y = \frac{3 - x}{2} \][/tex]
Next, let’s consider isolating [tex]\( y \)[/tex] from the second equation:
[tex]\[ 2x - 10y = 13 \][/tex]
We can isolate [tex]\( y \)[/tex] as follows:
[tex]\[ -10y = 13 - 2x \][/tex]
[tex]\[ y = \frac{13 - 2x}{-10} \][/tex]
[tex]\[ y = -\frac{13 - 2x}{10} \][/tex]
[tex]\[ y = -\frac{13}{10} + \frac{2x}{10} \][/tex]
[tex]\[ y = -\frac{13}{10} + \frac{x}{5} \][/tex]
We can isolate [tex]\( x \)[/tex] from the first equation:
[tex]\[ 3x + 6y = 9 \][/tex]
We can isolate [tex]\( x \)[/tex] as follows:
[tex]\[ 3x = 9 - 6y \][/tex]
[tex]\[ x = \frac{9 - 6y}{3} \][/tex]
[tex]\[ x = 3 - 2y \][/tex]
Or, we can isolate [tex]\( x \)[/tex] from the second equation:
[tex]\[ 2x - 10y = 13 \][/tex]
We can isolate [tex]\( x \)[/tex] as follows:
[tex]\[ 2x = 13 + 10y \][/tex]
[tex]\[ x = \frac{13 + 10y}{2} \][/tex]
Comparing all these isolated variables, we notice the simplest form is [tex]\( y \)[/tex] isolated from the first equation:
[tex]\[ y = \frac{3 - x}{2} \][/tex]
Hence, the best choice to substitute would be to isolate [tex]\( y \)[/tex] from the first equation:
(A) [tex]\( y \)[/tex], in the first equation.
