Question

A teacher wrote the equation 3y + 12 = 6x on the board. For what value of b would the additional equation 2y = 4x + b form a system of linear equations with infinitely many solutions?

b = –8
b = –4
b = 2
b = 6

Answer 1

To determine the value of \( b \) that would allow the equations \( 3y + 12 = 6x \) and \( 2y = 4x + b \) to have infinitely many solutions, we need to express both equations in the slope-intercept form \( y = mx + c \), where \( m \) is the slope.

First, let's rearrange the first equation:

1. Start with the equation:
\[
3y + 12 = 6x
\]

2. Subtract 12 from both sides:
\[
3y = 6x - 12
\]

3. Divide every term by 3:
\[
y = 2x - 4
\]

Now we have the first equation in slope-intercept form:
\[
y = 2x - 4
\]

Next, we'll rearrange the second equation:

1. Start with the equation:
\[
2y = 4x + b
\]

2. Divide every term by 2:
\[
y = 2x + \frac{b}{2}
\]

Now the second equation in slope-intercept form is:
\[
y = 2x + \frac{b}{2}
\]

For these two equations to represent the same line (and hence have infinitely many solutions), their slopes and intercepts must be equal:

1. The slopes are already equal (both are \( 2 \)).
2. Set the y-intercepts equal:
\[
-4 = \frac{b}{2}
\]

Now, multiply both sides by 2:
\[
-8 = b
\]

Now, let's check if the options given contain \( b = -8 \). Since they are:

- \( b = -8 \)
- \( b = -4 \)
- \( b = 2 \)
- \( b = 6 \)

The correct answer is:
\[
\boxed{-8}
\]