Match the description of the one variable equation with the number of solutions it will have.

Question

ax+c1=bx+c2
(3 points)
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The number of x's on both sides of the equation is different a≠b
The number of x's is the same on both sides of the equation, and the constant is different a=b, c1≠c2
The number of x's is the same on both sides of the equation and the constant is also the same a=b, c1=c2
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Answer 1

To match the descriptions with the number of solutions for the equation \( ax + c_1 = bx + c_2 \):

1. The number of x's on both sides of the equation is different \( a \neq b \):
   This will result in one solution since the terms can be combined and simplified to find a specific value for \( x \).

2. The number of x's is the same on both sides of the equation, and the constant is different \( a = b, c_1 \neq c_2 \):
   This will result in no solutions (the two sides of the equation will be parallel lines).

3. The number of x's is the same on both sides of the equation and the constant is also the same \( a = b, c_1 = c_2 \):
   This will result in infinite solutions (the two sides are identical, thus any x will satisfy the equation).

To summarize the matches based on the provided descriptions:

• \( a \neq b \) → One solution
• \( a = b, c_1 \neq c_2 \) → No solutions
• \( a = b, c_1 = c_2 \) → Infinite solutions