#16 · No solution? Same slope, different intercept.
The question. In the system below, k is a constant. The system has no solution. What is the value of k?
What "no solution" means. Each equation is a line, and a solution is a point that lies on both lines. No solution means the two lines never cross. So they're parallel: same slope, different y-intercepts.
The move: put both equations in the same shape. Move the x and y terms to the left in both equations, so each reads Ax + By = C. Subtract 3y from both sides of the first, and ky from both sides of the second:
The x terms already match: 6x in both. With the x terms equal, the lines have the same slope only if the y terms match too: −5y and −ky must be the same, so k = 5. (If the x terms don't match yet, scale one equation first; see below.)
Check the right sides. With k = 5, the left sides are identical, 6x − 5y, but one equals 4 and the other equals 14. The same expression can't equal both, so no x and y make both equations true. No solution, just as the question says.
When the x terms don't match yet
Scale one equation first. For 3x + 2y = 7 and 6x + ky = 5, multiply the first by 2 to get 6x + 4y = 14. Now the x terms match, so k = 4, and 14 isn't 5, so there's no solution. You can also check without rewriting: the x and y coefficients are in the same ratio (3 to 6, and 2 to 4), but the right sides aren't (7 to 5).
Check it in Desmos
Type both equations as written, with 5 in place of k. Desmos draws two parallel lines that never cross. But don't pick k by how the lines look: at k = 4.9 they look parallel too, yet they still cross, at (84, 100), outside the usual viewing window. Comparing the coefficients gives the exact value.
Going further: the same idea with one variable
A one-variable equation with no solution works the same way. 4(kx − 3) = 12x + 5 becomes 4kx − 12 = 12x + 5. Make the x terms match: 4k = 12, so k = 3. The numbers left over, −12 and 5, are different, so no value of x works.
The takeaway: no solution means parallel lines: same slope, different y-intercepts. Put both equations in the same shape, scale one if needed so the x terms match, then match the y terms and check that the right sides differ.