Final answer:
After finding the slope of the first equation and setting the second equation's slope equal to it, the correct value of K for the equations to have no solution is found to be -3, which corresponds to option A.
Step-by-step explanation:
To find the value of K when the equations 7x+14y−12=0 and Kx−5y+3=0 have no solutions, we first rewrite the given equations in slope-intercept form (y = mx + b) to compare their slopes and y-intercepts. Rewriting the first equation gives us y = −(1/2)x + (6/7). The slope of this line is −0.5. For two lines to have no solution, they must be parallel, meaning they must have the same slope but different y-intercepts.
Now, let's find the slope of the second equation, which we can derive by rearranging it to the form y = mx + b. We have y = (K/5)x − (3/5). Thus, the ratio K/5 must equal to −0.5 for the lines to be parallel and have no solutions. Setting K/5 equal to −0.5 gives us K = −2.5 or simply K = −3 when we multiply both sides by 5. Therefore, option A, K = −3, is the correct answer.