Final answer:
After simplifying and solving each equation, the first has no solution, while the second and third have one solution each.
Step-by-step explanation:
We are tasked with dragging the number of solutions to match each given equation. Let's analyze each equation step by step to determine the number of solutions.
Solution Steps:
- Simplify the equations by expanding and combining like terms.
- Rearrange the equation to solve for x.
- Determine the number of solutions based on the simplified form of the equation.
The first equation, 4(3x + 2) = 4(2x + 3) + 4x, simplifies to 12x + 8 = 8x + 12 + 4x. Combining like terms gives us 12x + 8 = 12x + 12. Subtracting 12x from both sides, we get 8 = 12, which is untrue. Therefore, this equation has no solution.
The second equation, 10(x + 4) – 3 = 14x + 1, simplifies to 10x + 40 – 3 = 14x + 1. Combining like terms gives us 10x + 37 = 14x + 1. Rearranging to solve for x, we subtract 10x from both sides and get 37 = 4x + 1. Subtracting 1 from both sides gives us 36 = 4x, and dividing by 4, we get x = 9. This equation has one solution.
The third equation, (10x + 15) – ½ = 2x + 6 + 3x, simplifies to 10x + 14.5 = 5x + 6. Subtracting 5x from both sides, we get 5x + 14.5 = 6. Subtracting 14.5 from both sides gives us 5x = -8.5, and dividing by 5, we get x = -1.7. This equation has one solution.
As a result, the correct order of the number of solutions corresponding to the equations given is:
- No solution
- One solution
- One solution