Question

For each of the following system of linear equations, state the number of solutions without solving the system.

a. − + 3 = 9, −4 + 12 = 12
b. 2x - y - 4 = 0, 6x = 3y + 12

Answer 1

For the given systems of linear equations, both sets a and b have infinitely many solutions because the equations in each set are multiples of each other, indicating that the lines are coincident.

Step-by-step explanation:

For each of the following systems of linear equations, we can determine the number of solutions without solving the systems by comparing the coefficients of the variables in each equation. Let's examine each set of equations:

• a. −x + 3y = 9, −4x + 12y = 12
• b. 2x - y - 4 = 0, 6x = 3y + 12

For the first set (a), if we divide the second equation by −4, we get x - 3y = −3, which is a multiple of the first equation. This implies that the two lines are coincident, and thus there are infinitely many solutions since the lines are on top of each other.

For the second set (b), if we manipulate the second equation by dividing by 3, we get 2x = y + 4, which when rearranged is 2x - y - 4 = 0, identical to the first equation. This indicates, as with the first set, that the lines are coincident, suggesting infinitely many solutions as well.