To determine the value of [tex]\(b\)[/tex] that will cause the system to have an infinite number of solutions, we need to make the two equations equivalent. The given system of equations is:
[tex]\[ y = 6x - b \][/tex]
[tex]\[ -3x + \frac{1}{2}y = -3 \][/tex]
First, let's simplify the second equation by eliminating the fraction. Multiply the entire equation by 2:
[tex]\[ -3x + \frac{1}{2}y = -3 \][/tex]
[tex]\[ 2(-3x + \frac{1}{2}y) = 2(-3) \][/tex]
[tex]\[ -6x + y = -6 \][/tex]
Now, we have the following system of equations:
[tex]\[ y = 6x - b \][/tex]
[tex]\[ y = 6x - 6 \][/tex] (after rearranging the simplified second equation to match the first equation's form)
For the two equations to be equivalent, their right-hand sides must be the same:
[tex]\[ 6x - b = 6x - 6 \][/tex]
Subtracting [tex]\(6x\)[/tex] from both sides of the equation, we get:
[tex]\[ -b = -6 \][/tex]
To solve for [tex]\(b\)[/tex], multiply both sides of the equation by -1:
[tex]\[ b = 6 \][/tex]
Thus, the value of [tex]\(b\)[/tex] that will cause the system to have an infinite number of solutions is:
[tex]\[ \boxed{6} \][/tex]
