If the system of the linear equations above has infinitely many solutions, and c is a constant, what is the value of c? a) - 6b) - 3c) - 2d) - 1

1. For the system of linear equations to have an infinite number of solutions, the two lines must cross over an infinite number of times, ie. they would be the same line.

Now, if they are the same line and we were to write out both equations in the form y = mx + c, this would mean that the m value (the gradient) and the c value (the y-intercept) would be the same. Thus, the first step is to write out both equations in the form y = mx + c.

a) Equation 1:

3x - 9y = -6

3x = -6 + 9y (Add 9y to both sides)

3x + 6 = 9y (Add 6 to both sides)

(1/3)x + 2/3 = y (Divide both sides by 9)

Thus, the first equation may be written as y = (1/3)x + 2/3

b) Equation 2:

(1/2)x - (3/2)y = c

(1/2)x = c + (3/2)y (Add (3/2)y to both sides)

(1/2)x - c = (3/2)y (Subtract c from both sides)

(1/3)x - (2/3)c = y (Multiply both sides by 2/3)

Thus, the second equation may be written as y = (1/3)x - (2/3)c.

2. Now that we have both equations in the form y = mx + c, we can look at their gradients and y-intercepts:

Equation 1: gradient = 1/3, y-intercept = 2/3

Equation 2: gradient = 1/3, y-intercept = -(2/3)c

As we can see above, the gradients of the two lines are the same, however for them to be the same line their y-intercepts must also be equal. Thus, we must equate the two y-intercepts to find c:

2/3 = -(2/3)c

1 = -c (Multiply both sides by 3/2)

-1 = c (Multiply both sides by -1)

Thus, if the system of linear equations has infinitely many solutions, and c is a constant, the value of c is -1 (answer D).

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