Solve the following system of linear equations using the substitution method: \[\begin{align*} x + 2y &= 7 \\ 3x - y &= 5 \end{align*}\]
Answer
Simplify to find the value of \(x\): \[x = 7 - \frac{32}{7}\] or \[x = \frac{49}{7} - \frac{32}{7} = \frac{17}{7}\]
Step 1 ：First, isolate \(x\) in the first equation: \[x = 7 - 2y\]
Step 2 ：Then substitute \(x\) in the second equation: \[3(7 - 2y) - y = 5\]
Step 3 ：Simplify the equation: \[21 - 6y - y = 5\] or \[21 - 7y = 5\]
Step 4 ：Rearrange the equation to solve for \(y\): \[7y = 21 - 5\] or \[y = \frac{16}{7}\]
Step 5 ：Substitute \(y\) back into the first equation to solve for \(x\): \[x = 7 - 2 \left(\frac{16}{7}\right)\]
Step 6 ：Simplify to find the value of \(x\): \[x = 7 - \frac{32}{7}\] or \[x = \frac{49}{7} - \frac{32}{7} = \frac{17}{7}\]
