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3 of 53 of 5 Items Question Match the description of the one variable equation with the number of solutions it will have. ax+c1=bx+c2 (3 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. 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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To match the descriptions of the one-variable equation \( ax + c_1 = bx + c_2 \) with the number of solutions it will have, the breakdown is as follows:

  1. The number of x's on both sides of the equation is different: This scenario occurs when \( a \neq b \). In this case, the equation will have one unique solution.

  2. The number of x's is the same on both sides of the equation, and the constant is different: This happens when \( a = b \) and \( c_1 \neq c_2 \). In this case, the equation will have no solution (inconsistent).

  3. The number of x's is the same on both sides of the equation and the constant is also the same: This situation occurs when \( a = b \) and \( c_1 = c_2 \). In this case, the equation will have infinitely many solutions (consistent).

So, the answers are:

  1. Different number of x's: \( a \neq b \) - One unique solution.
  2. Same number of x's, different constants: \( a = b, c_1 \neq c_2 \) - No solution.
  3. Same number of x's, same constants: \( a = b, c_1 = c_2 \) - Infinitely many solutions.