Rewrite every line into this form before comparing two lines.
When does a system have no / one / infinite solutions?
No solution → same slope, different intercept (parallel lines).
Infinitely many → same slope AND same intercept (the same line).
Exactly one → different slopes.
So "no solution" questions are really "make the slopes equal" questions.
Translating a line
Down k units: replace y with (y + k), i.e. subtract k from the whole right side of y = …
Shortcut in standard form Ax + By = C: shifting down k changes C by −Bk.
To find an x-intercept, set y = 0 and solve for x.
Slope from two points
m = (y₂ − y₁) / (x₂ − x₁)
Plug the two given points in carefully — keep the order consistent top and bottom.
Worked example — translate then find the x-intercept
Problem. The graph of 9x − 10y = 19 is translated down 4 units. Find the x-coordinate of the x-intercept.
Down 4 units: y → y + 4, so 9x − 10(y + 4) = 19.
Expand: 9x − 10y − 40 = 19 → 9x − 10y = 59.
x-intercept: set y = 0 → 9x = 59 → x = 59/9 ≈ 6.556.
Watch out: Translating down k means y becomes (y + k), which raises the constant on the other side. Getting the sign backwards (using y − k) is exactly how M1 Q22 was missed. Also: M1 Q9 asked for x − 1, not x — read the ask.
Video lessons
Curated walkthroughs from trusted math channels — watch one or two before doing the worksheet.