Final answer:
To determine if an equation has one solution, no solutions, or infinite solutions, analyze the equation's variables and constants. Equations are solved by identifying unknowns and knowns, selecting an appropriate equation, and performing algebraic operations to isolate the variable.
Step-by-step explanation:
To determine if an equation has one solution, no solutions, or infinite solutions, we need to analyze the equation after applying algebraic methods. To illustrate, let's consider a linear equation in the form ax + b = cx + d. To find the solution, we subtract cx from both sides to get (a - c)x = d - b.
If a - c ≠ 0, the equation will have exactly one solution, which can be found by dividing both sides by (a - c). If a - c = 0 and d - b ≠ 0, the equation will have no solutions, as we would end up with a contradiction (e.g., 0x = 5, which is false). If a - c = 0 and d - b = 0, the equation will have infinite solutions, since we would have 0x = 0, which is true for all values of x.
To solve for the unknown variable, we:
1. Identify the unknown.
2. Identify the knowns.
3. Choose an equation, plug in the knowns, and solve for the unknown.
Let's apply this method with an example equation: 2x + 3 = 4x - 1. First, we subtract 2x from both sides to get 3 = 2x - 1. Next, we add 1 to both sides to obtain 4 = 2x. Finally, we divide by 2 to find x = 2. This example has one solution: x=2.
In summary, to solve an equation, we need to identify the unknowns and the knowns, select the appropriate equation(s), and algebraically manipulate the equation to solve for the unknown.