In a linear system, infinitely many solutions means the equations draw the same line. Find the multiplier that takes both variable terms in the first equation to those in the second, then apply it to the constant too. Desmos can confirm that the lines overlap, but comparing the terms is faster here. Scaling only the variable terms would produce parallel lines instead.
Hints
- Hint 1
Infinitely many solutions means every point on one line is also on the other. What number changes into ? Check whether that same number changes into .
- Hint 2
A multiplier must act on the whole equation, not only the terms with variables. Once you know how the left side changes, what must happen to the on the right side?
Step-by-step
Match the whole equation
Step 1Find the multiplier
A solution is a pair that satisfies both equations. For infinitely many pairs to work, the equations must draw the same line. The term changes from to by a factor of , and the term follows that factor too:
- Step 2
Scale the constant too
Multiply both sides by the same factor to keep the same line. The second equation's left side is triple the first, so its right side must be triple the :
Leaving the constant at would give a different, parallel line.
- Step 3
Read the value and check the lines
Type in Desmos, followed by and . Desmos shows , and the graphs lie on top of each other, so they share infinitely many points. The value of is . Choice C.