Final answer:
The value of “a” that makes the equation 12x – 5 = 3(ax – 7) have no solutions is 4. This is because it results in two parallel lines with different y-intercepts.
Step-by-step explanation:
The student is asking what value of a would make the given linear equation have no solutions. The original equation is 12x – 5 = 3(ax – 7). To have no solution, the lines represented by both sides of the equation must be parallel, which means their slopes must be equal, but the y-intercepts must be different. We can find the value of a by equating the coefficients of x from the left and right side of the equation, as long as the constants (y-intercepts) on each side do not become equal after simplifying.
First, distribute the 3 on the right side to get 12x – 5 = 3ax - 21. The coefficient of x on the left side is 12, so setting 3a equal to 12 gives us a = 4. However, if a is 4, then the constant terms on both sides will not match, as we have – 5 on the left side and – 21 on the right side. This means that when a is 4, we end up with two parallel lines that have different y-intercepts, hence there are no solutions to the equation.
To conclude, the value a = 4 makes the equation have no solutions.