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What value of [tex]\( b \)[/tex] will cause the system to have an infinite number of solutions?

[tex]\[
\begin{array}{l}
y = 6x - b \\
-3x + \frac{1}{2}y = -3
\end{array}
\][/tex]

[tex]\( b = \)[/tex]

A. 2
B. 4
C. 6
D. 8



Answer :

GinnyAnswer
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]

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