Questions: What value of b will cause the system to have an infinite number of solutions? y=6 x+b -3 x+1/2 y=-3

Transcript text: What value of $b$ will cause the system to have an infinite number of solutions? \[ \begin{array}{l} y=6 x+b \\ -3 x+\frac{1}{2} y=-3 \end{array} \]

Solution

Solution Steps

Step 1: Rewrite the second equation in slope-intercept form.

Add $3x$ to both sides of the second equation: $\frac{1}{2}y = 3x - 3$

Multiply both sides by 2: $y = 6x - 6$

Step 2: Compare the two equations.

The first equation is $y = 6x + b$. The second equation is $y = 6x - 6$. For the system to have infinitely many solutions, the two equations must represent the same line. This means the two equations must be identical.

Step 3: Find the value of $b$.

Comparing the two equations, we see that they have the same slope, which is 6. For the lines to be identical, they must also have the same y-intercept. Thus, $b$ must be equal to -6.