In a linear system, infinitely many solutions means the equations draw the same line. Compare the and coefficients to find the multiplier between their left sides, then apply it to the constant as well. You can graph both equations in Desmos to confirm they overlap. Matching coefficients without matching constants gives parallel lines instead.
Hints
- Hint 1
For infinitely many solutions, every point on one line must also be on the other. What number turns both into and into ?
- Hint 2
Multiplying an equation by a number keeps the same line only if you multiply both sides. Once you've found the multiplier for the left side, what must happen to the constant on the right?
Step-by-step
Match the whole equation
Step 1Find the multiplier between the left sides
The term goes from to , and the term goes from to . Both are doubled, so
Infinitely many solutions means the two equations describe the same line. Matching the left sides is only the first check.
- Step 2
Apply the multiplier to the constant
Multiply both sides by the same number to keep the first equation true at all the same points. Multiply the first equation by :
The doubled left side matches the second equation, so its right side must match the doubled constant too. Leaving the constant at would give a different, parallel line.
- Step 3
Confirm the lines overlap
Type , , and on separate lines in Desmos. It shows , and the graphs lie on top of each other. Every point on that line satisfies both equations, so the value of is . Choice C.