The question in the image asks the following:

"Isaac wrote two simplified expressions that were not equal to each other. Each equation also had a different coefficient on the variable. If he sets the expressions equal to each other, will the equation have one solution, no solution, or infinitely many solutions? Use an example to support your answer."

Explanation:

When two expressions with different coefficients on the variable are set equal to each other, the equation typically has one solution. This is because different coefficients lead to a situation where the expressions will intersect at exactly one point, meaning there is a unique value of the variable that satisfies the equation.

Example:

Consider two expressions:

1. $2x + 3$
2. $5x - 1$

If we set them equal to each other: $2x + 3 = 5x - 1$

Solving for $x$:

1. Subtract $2x$ from both sides: $3 = 3x - 1$

2. Add 1 to both sides: $4 = 3x$

3. Divide by 3: $x = \frac{4}{3}$

Thus, the equation has one solution: $x = \frac{4}{3}$.

Conclusion:

If the coefficients are different, the equation will typically have one solution. If the coefficients and constants were the same, it would have infinitely many solutions, but if only the constants were different with identical coefficients, there would be no solution.

Do you want any further details or have any questions?

Here are 5 related questions for deeper exploration:

1. What happens if both expressions have the same coefficient but different constants?
2. Can an equation have no solutions if the expressions are entirely different?
3. How would the solution change if one coefficient is negative?
4. What is the geometric interpretation of a system with one solution?
5. How can we identify equations with infinitely many solutions or no solutions quickly?

Tip: Always simplify both sides of an equation before solving to avoid errors in handling terms.