Answer :
To solve the given system of equations, we follow these steps:
[tex]\[ \begin{array}{l} 1. \, y = \frac{1}{3} x + 5 \\ 2. \, y = 2x \end{array} \][/tex]
Step 1: Set the two expressions for [tex]\( y \)[/tex] equal to each other, since both equations describe [tex]\( y \)[/tex].
[tex]\[ \frac{1}{3} x + 5 = 2x \][/tex]
Step 2: Solve the resulting equation for [tex]\( x \)[/tex].
First, eliminate the fraction by multiplying every term by 3:
[tex]\[ 3 \left( \frac{1}{3} x \right) + 3 \cdot 5 = 3 \cdot 2x \][/tex]
Simplifying:
[tex]\[ x + 15 = 6x \][/tex]
Next, isolate [tex]\( x \)[/tex] by subtracting [tex]\( x \)[/tex] from both sides:
[tex]\[ 15 = 6x - x \][/tex]
[tex]\[ 15 = 5x \][/tex]
Finally, solve for [tex]\( x \)[/tex] by dividing both sides by 5:
[tex]\[ x = 3 \][/tex]
Step 3: Substitute [tex]\( x = 3 \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]. Using the second equation [tex]\( y = 2x \)[/tex]:
[tex]\[ y = 2 \cdot 3 \][/tex]
[tex]\[ y = 6 \][/tex]
Step 4: Find the value of [tex]\( x + y \)[/tex].
[tex]\[ x + y = 3 + 6 = 9 \][/tex]
Thus, the value of [tex]\( x + y \)[/tex] is:
[tex]\[ \boxed{9} \][/tex]
[tex]\[ \begin{array}{l} 1. \, y = \frac{1}{3} x + 5 \\ 2. \, y = 2x \end{array} \][/tex]
Step 1: Set the two expressions for [tex]\( y \)[/tex] equal to each other, since both equations describe [tex]\( y \)[/tex].
[tex]\[ \frac{1}{3} x + 5 = 2x \][/tex]
Step 2: Solve the resulting equation for [tex]\( x \)[/tex].
First, eliminate the fraction by multiplying every term by 3:
[tex]\[ 3 \left( \frac{1}{3} x \right) + 3 \cdot 5 = 3 \cdot 2x \][/tex]
Simplifying:
[tex]\[ x + 15 = 6x \][/tex]
Next, isolate [tex]\( x \)[/tex] by subtracting [tex]\( x \)[/tex] from both sides:
[tex]\[ 15 = 6x - x \][/tex]
[tex]\[ 15 = 5x \][/tex]
Finally, solve for [tex]\( x \)[/tex] by dividing both sides by 5:
[tex]\[ x = 3 \][/tex]
Step 3: Substitute [tex]\( x = 3 \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]. Using the second equation [tex]\( y = 2x \)[/tex]:
[tex]\[ y = 2 \cdot 3 \][/tex]
[tex]\[ y = 6 \][/tex]
Step 4: Find the value of [tex]\( x + y \)[/tex].
[tex]\[ x + y = 3 + 6 = 9 \][/tex]
Thus, the value of [tex]\( x + y \)[/tex] is:
[tex]\[ \boxed{9} \][/tex]