Question

analyzing a system of linear equations
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?
( circ b = -8 )
( circ b = -4 )
( circ b = 2 )
( circ b = 6 )

Explanation:

Step1: Simplify first equation

Rearrange $3y + 12 = 6x$ to slope-intercept form:
Subtract 12 from both sides: $3y = 6x - 12$
Divide by 3: $y = 2x - 4$

Step2: Simplify second equation

Rearrange $2y = 4x + b$ to slope-intercept form:
Divide by 2: $y = 2x + \frac{b}{2}$

Step3: Set constants equal for infinite solutions

For infinitely many solutions, the equations must be identical (same slope and y-intercept). Set the y-intercepts equal:
$\frac{b}{2} = -4$

Step4: Solve for b

Multiply both sides by 2:
$b = -8$

Answer:

A. $b = -8$