What value of b will cause the system to have an infinite number of solutions?

Question

A system of equations. y equals 6 x plus b. negative 3 x plus StartFraction one-half EndFraction y equals negative 3.

A coordinate grid with a line labeled negative 3 x plus StartFraction one-half EndFraction y equals negative 3 and passes through the points (1, 0) and (0, negative 6).

-6
-3
3
6

Answer 1

To determine the value of \( b \) that will cause the system of equations to have an infinite number of solutions, we need to analyze the given equations:

1. The first equation is given as: \[ y = 6x + b \]

2. The second equation can be re-arranged. Given: \[ -3x + \frac{1}{2}y = -3 \] we can multiply all terms by 2 to eliminate the fraction: \[ -6x + y = -6 \] Rearranging this gives us: \[ y = 6x - 6 \]

For the system to have an infinite number of solutions, the two equations must represent the same line. This will happen if their slopes and y-intercepts are the same.

From the equations, we can see that:

• The slope of the first equation \( y = 6x + b \) is \( 6 \) and its y-intercept is \( b \).
• The slope of the second equation \( y = 6x - 6 \) is also \( 6 \) and its y-intercept is \(-6\).

For the two lines to be the same, the y-intercepts must equal each other. Therefore, we set \( b \) equal to \(-6\): \[ b = -6 \]

Thus, the value of \( b \) that will cause the system to have an infinite number of solutions is: \[ \boxed{-6} \]