Question

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

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 of the one-variable equation \( ax + c_1 = bx + c_2 \) with the number of solutions, we can break down each scenario:

1. **The number of x's on both sides of the equation is different \( a \neq b \)**:
- In this case, since the coefficients of \( x \) are different, there will be exactly one solution for \( x \).

2. **The number of x's is the same on both sides of the equation, and the constant is different \( a = b, c_1 \neq c_2 \)**:
- Here, since the coefficients are the same, but the constants are different, the equation simplifies to \( 0 = c_2 - c_1 \), which is a contradiction. Therefore, there are no solutions.

3. **The number of x's is the same on both sides of the equation and the constant is also the same \( a = b, c_1 = c_2 \)**:
- In this scenario, since both the coefficients of \( x \) and the constants are the same, the equation simplifies to \( 0 = 0 \), which is always true. Therefore, there are infinitely many solutions.

Now to summarize:

- \( a \neq b \) → **One solution**
- \( a = b, c_1 \neq c_2 \) → **No solution**
- \( a = b, c_1 = c_2 \) → **Infinitely many solutions**

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