Question

$$ -3 x+2=p(x-q) $$ In the equation above, \(p\) and \(q\) are constants. If there are infinitely many solutions to the equation, what is the value of \(q\) ?

Short answer

The value of \(q\) such that there are infinitely many solutions to the given equation is \(-\frac{2}{3}\).

Step-by-step solution

Rewrite the equation in standard form

To rewrite the equation in standard form, we want to get all terms on one side of the equation and set the equation to zero. So, we will expand the right side, then move all terms to the left side of the equation:\[ -3x + 2 = p(x - q) \implies -3x + 2 = px - pq \] Next, shift all terms to the left side of the equation:\[ -3x + 2 - px + pq =0 \]

Identify conditions for infinitely many solutions

For the equation to have infinitely many solutions, the coefficients of x terms must be equal, and the constant terms must also be equal. This implies the following conditions:\[ -3 = p \] and \[ 2 = pq \]

Solve for q

Now we can use these two conditions to find the value of q. Since \(-3 = p\), we can substitute this value into the equation for the constant terms:\[ 2 = (-3)q \] Now, divide both sides by \(-3\) to find the value of \(q\):\[ q = \frac{2}{-3} \] So, the value of \(q\) such that there are infinitely many solutions to the given equation is \(-\frac{2}{3}\).