Certainly! Let's solve the system of linear equations step by step:
We are given two equations:
1. [tex]\( x + 2y = 7 \)[/tex]
2. [tex]\( x = \frac{y}{2} \)[/tex]
### Step 1: Isolate [tex]\( x \)[/tex] from the second equation
The second equation already expresses [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ x = \frac{y}{2} \][/tex]
### Step 2: Substitute [tex]\( x \)[/tex] into the first equation
Substitute [tex]\( x = \frac{y}{2} \)[/tex] into the first equation:
[tex]\[ \frac{y}{2} + 2y = 7 \][/tex]
### Step 3: Combine like terms
Combine the terms involving [tex]\( y \)[/tex] on the left-hand side:
[tex]\[ \frac{y}{2} + 2y = \frac{y}{2} + \frac{4y}{2} = \frac{5y}{2} \][/tex]
So, the equation becomes:
[tex]\[ \frac{5y}{2} = 7 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
To solve for [tex]\( y \)[/tex], multiply both sides of the equation by 2:
[tex]\[ 5y = 14 \][/tex]
Now, divide both sides by 5:
[tex]\[ y = \frac{14}{5} \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Now that we know [tex]\( y = \frac{14}{5} \)[/tex], we can find [tex]\( x \)[/tex] using the second equation [tex]\( x = \frac{y}{2} \)[/tex]:
[tex]\[ x = \frac{\frac{14}{5}}{2} = \frac{14}{5} \cdot \frac{1}{2} = \frac{14}{10} = \frac{7}{5} \][/tex]
### Step 6: Write the solution
Thus, the solution to the system of equations is:
[tex]\[ x = \frac{7}{5}, \quad y = \frac{14}{5} \][/tex]
