In a system of two linear equations, “infinitely many solutions” means both equations draw the same line. Compare the variable coefficients to find the multiplier, then apply it to the constant. You can confirm the lines overlap in Desmos. Scaling only the variable terms gives parallel lines, not infinitely many solutions.
Hints
- Hint 1
A solution is a point that satisfies both equations. To have infinitely many solutions, the equations must describe the same line. What would make one whole equation a copy of the other?
- Hint 2
Compare the coefficients: what number takes to ? Check whether that same number also takes to .
- Hint 3
Multiplying both sides of an equation by the same nonzero number keeps its solutions unchanged. Once the variable terms match, apply that same multiplier to the constant .
Step-by-step
Match the whole equation
Step 1Find the multiplier
A solution satisfies both equations. For infinitely many solutions, the equations must describe the same line. Find the multiplier that takes the first coefficient, , to the second, :
- Step 2
Check the other coefficient
Check that the same multiplier works on the coefficient:
That matches the second equation. Both sides of an equation must scale together for the lines to be identical. Leaving the constant unchanged would give parallel lines instead.
- Step 3
Scale the constant and confirm
Type in Desmos to scale the constant. Then type both equations. Desmos shows , and the graphs overlap: every point on that line satisfies both equations. So the system has infinitely many solutions when . Choice D.