Question

Which statement about the equation 3 (4x - 10) = gr - 30 + 2x is true?

O lf g = 10, the equation has infinitely many solutions.
O If g=10, the equation has no solution.
O lf g = 12, the equation has infinitely many solutions.
O lf g = 12, the equation has no solution.

Answer 1

When g=10, the equation simplifies to a form that shows the variables x and r are directly proportional, suggesting infinitely many solutions; similarly, when g=12, the equation also simplifies to show a direct proportionality, meaning there are infinitely many solutions. Option 1 is the correct answer.

Step-by-step explanation:

The given equation is 3 (4x - 10) = gr - 30 + 2x. To determine the truth of the statements regarding the values of g, we must first simplify the equation. Let's solve for x when g is replaced by both 10 and 12, respectively.

For g=10, the equation simplifies to 12x - 30 = 10r - 30 + 2x. This simplifies further to 10x = 10r, which means x = r for any value of r. This indicates that there are infinitely many solutions when g equals 10, as the variables x and r are directly proportional.

For g=12, the equation simplifies to 12x - 30 = 12r - 30 + 2x. This reduces to 10x = 12r, which implies that x = 1.2r. Again, this suggests there are infinitely many solutions because for any value of r, there's a corresponding value of x.

The correct option is: If g = 10, the equation has infinitely many solutions.