Certainly! Let’s solve the given equations step by step.
We have the system of equations:
[tex]\[ 2a + b = 8 \tag{1} \][/tex]
[tex]\[ a - b = 3 \tag{2} \][/tex]
First, we will solve this system of equations for [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
### Step 1: Solve Equation (2) for [tex]\( a \)[/tex]
From Equation (2):
[tex]\[ a - b = 3 \][/tex]
we can express [tex]\( a \)[/tex] in terms of [tex]\( b \)[/tex]:
[tex]\[ a = b + 3 \tag{3} \][/tex]
### Step 2: Substitute Equation (3) into Equation (1)
Now, substitute [tex]\( a = b + 3 \)[/tex] into Equation (1):
[tex]\[ 2(b + 3) + b = 8 \][/tex]
Simplify and solve for [tex]\( b \)[/tex]:
[tex]\[ 2b + 6 + b = 8 \][/tex]
[tex]\[ 3b + 6 = 8 \][/tex]
[tex]\[ 3b = 2 \][/tex]
[tex]\[ b = \frac{2}{3} \][/tex]
### Step 3: Find [tex]\( a \)[/tex]
Now, substitute [tex]\( b = \frac{2}{3} \)[/tex] into Equation (3):
[tex]\[ a = \left( \frac{2}{3} \right) + 3 \][/tex]
[tex]\[ a = \frac{2}{3} + \frac{9}{3} \][/tex]
[tex]\[ a = \frac{11}{3} \][/tex]
### Step 4: Calculate [tex]\( 5a + b \)[/tex]
Now that we have [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we can find [tex]\( 5a + b \)[/tex]:
[tex]\[ 5a + b = 5 \left( \frac{11}{3} \right) + \frac{2}{3} \][/tex]
[tex]\[ 5a + b = \frac{55}{3} + \frac{2}{3} \][/tex]
[tex]\[ 5a + b = \frac{57}{3} \][/tex]
[tex]\[ 5a + b = 19 \][/tex]
Therefore, the value of [tex]\( 5a + b \)[/tex] is [tex]\( 19 \)[/tex].
