Let's solve the given system of equations step-by-step to find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex], and then use these values to determine [tex]\( 5a + b \)[/tex].
We start with the two given equations:
[tex]\[ 2a + b = 8 \][/tex] [tex]\[ (1) \][/tex]
[tex]\[ a - b = 3 \][/tex] [tex]\[ (2) \][/tex]
Step 1: Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
From equation [tex]\( (2) \)[/tex]:
[tex]\[ a - b = 3 \][/tex]
We can solve for [tex]\( a \)[/tex]:
[tex]\[ a = b + 3 \][/tex] [tex]\( (3) \)[/tex]
Now we substitute equation [tex]\( (3) \)[/tex] into equation [tex]\( (1) \)[/tex]:
[tex]\[ 2(b + 3) + b = 8 \][/tex]
Expanding and simplifying:
[tex]\[ 2b + 6 + b = 8 \][/tex]
[tex]\[ 3b + 6 = 8 \][/tex]
Subtract 6 from both sides:
[tex]\[ 3b = 2 \][/tex]
Divide by 3:
[tex]\[ b = \frac{2}{3} \][/tex]
Now, we substitute [tex]\( b = \frac{2}{3} \)[/tex] back into equation [tex]\( (3) \)[/tex]:
[tex]\[ a = \frac{2}{3} + 3 \][/tex]
[tex]\[ a = \frac{2}{3} + \frac{9}{3} \][/tex]
[tex]\[ a = \frac{11}{3} \][/tex]
So we have:
[tex]\[ a = \frac{11}{3} \][/tex]
[tex]\[ b = \frac{2}{3} \][/tex]
Step 2: Calculate [tex]\( 5a + b \)[/tex]:
Now, using the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ 5a + b = 5 \left( \frac{11}{3} \right) + \frac{2}{3} \][/tex]
First, multiply [tex]\( 5 \times \frac{11}{3} \)[/tex]:
[tex]\[ 5 \times \frac{11}{3} = \frac{55}{3} \][/tex]
Now, add [tex]\(\frac{2}{3} \)[/tex]:
[tex]\[ 5a + b = \frac{55}{3} + \frac{2}{3} \][/tex]
[tex]\[ 5a + b = \frac{57}{3} \][/tex]
Simplify the fraction:
[tex]\[ 5a + b = 19 \][/tex]
Therefore, the value of [tex]\( 5a + b \)[/tex] is [tex]\( 19 \)[/tex].
