Alina solved this equation using the steps shown in the diagram, but she wasn't sure how to match this card.

What advice can you offer Alina for how to determine whether this equation has one solution, no solution, or infinitely many solutions?

To help Alina determine whether an equation has one solution, no solution, or infinitely many solutions, she can follow these steps:

1. **Simplify Both Sides of the Equation:** Start by simplifying the equation as much as possible. Combine like terms and distribute any factors if needed.

2. **Compare Coefficients and Constants:** After simplifying, focus on the coefficients of the variables and the constants (the numbers without variables). This comparison will help in determining the type of solution.

   - **One Solution:** If the coefficients of the variables are different, the equation has one solution. This happens because the two sides of the equation will intersect at exactly one point on a graph.
   - **No Solution:** If the coefficients of the variables are the same but the constants are different, the equation has no solution. This occurs because the two sides are parallel lines that never intersect.
   - **Infinitely Many Solutions:** If both the coefficients of the variables and the constants are the same, the equation has infinitely many solutions. In this case, the two sides of the equation are identical, representing the same line.

3. **Solve the Equation (if needed):** If the equation isn't immediately clear, Alina can solve for the variable to see the result. If she gets a specific value (like \( x = 3 \)), it has one solution. If she gets a false statement (like \( 5 = 3 \)), there is no solution. If she gets a true statement (like \( 0 = 0 \)), there are infinitely many solutions.

By following these steps, Alina will be able to categorize the equation correctly.

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**Relative Questions:**

1. What happens if the equation includes fractions or decimals?
2. How do you handle equations with variables on both sides?
3. Can you provide an example where an equation has no solution?
4. What does it mean graphically when an equation has infinitely many solutions?
5. How do equations with different exponents affect the number of solutions?

**Tip:** Always double-check your work by plugging the solution back into the original equation to ensure it satisfies all conditions.

Understand how to determine if an equation has one solution, no solution, or infinitely many solutions with detailed steps and examples. This guide covers key algebraic concepts, including coefficients, constants, and solving equations. Suitable for students in Grades 7-9.

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