A teacher wrote the equation [tex]$3 y+12=6 x$[/tex] on the board. For what value of b would the additional equation [tex]$2 y=4 x$[/tex] [tex]$b$[/tex] form a system of linear equations with infinitely many solutions?
A. [tex]$b=-8$[/tex]
B. [tex]$b=-4$[/tex]
C. [tex]$b=2$[/tex]
D. [tex]$b=6$[/tex]

Question

GinnyAnswer · Answer

Rewrite both equations in slope-intercept form. 
 Compare the slopes and y-intercepts of the two equations. 
 Set the y-intercepts equal to each other: − 4 = 2 b ​ . 
 Solve for b: b = − 8 , so the final answer is − 8 ​ . 
 
 Explanation 
 
 Analyze the problem and available data We are given two linear equations: 
 
 Equation 1: 3 y + 12 = 6 x Equation 2: 2 y = 4 x + b 
 We want to find the value of b such that the system of equations has infinitely many solutions. This occurs when the two equations represent the same line. To determine this, we will rewrite both equations in slope-intercept form ( y = m x + c ), where m is the slope and c is the y-intercept. 
 
 Rewrite Equation 1 First, let's rewrite Equation 1 in slope-intercept form: 
 
 3 y + 12 = 6 x 3 y = 6 x − 12 y = 3 6 x − 12 ​ y = 2 x − 4 
 
 Rewrite Equation 2 Now, let's rewrite Equation 2 in slope-intercept form: 
 
 2 y = 4 x + b y = 2 4 x + b ​ y = 2 x + 2 b ​ 
 
 Compare slopes and y-intercepts For the system to have infinitely many solutions, the two lines must be identical. This means they must have the same slope and the same y-intercept. 
 
 Comparing the two equations in slope-intercept form: 
 Equation 1: y = 2 x − 4 Equation 2: y = 2 x + 2 b ​ 
 The slopes are both 2, so they are equal. Now we need to make the y-intercepts equal: 
 − 4 = 2 b ​ 
 
 Solve for b To solve for b , we multiply both sides of the equation by 2: 
 
 2 × − 4 = 2 × 2 b ​ − 8 = b 
 Therefore, b = − 8 . 
 
 Final Answer The value of b for which the system of equations has infinitely many solutions is -8. 
 
 Examples 
 In economics, consider two supply equations for a product. If these equations are linearly dependent (representing the same supply curve), it means the market supply is consistent regardless of which equation you use. This situation, analogous to infinitely many solutions, helps in simplifying market models and understanding supply-demand equilibrium.

Anonymous · Answer

The value of b for which the equations have infinitely many solutions is − 8 . Therefore, the correct answer is A. b = − 8 . 
 ;

Answer (2)

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