Final answer:
To determine the number of solutions to an equation, identify the unknown and known values, select the appropriate equation(s), and solve for the unknown. A single solution exists when the variable can be isolated with a real number result; infinite solutions arise when the equation simplifies to a true statement like 0=0 for all values of the variable; no solutions occur when the equation simplifies to an impossible statement, such as 0=5.
Step-by-step explanation:
To determine if the equation has one solution, no solutions, or infinite solutions, first we need to identify the unknown variable and the known values or constants. Then, we choose an appropriate equation or set of equations to solve for the unknown. Solving the equation involves algebraic manipulation, which might include simplifying terms, combining like terms, or using methods such as substitution or elimination if working with systems of equations.
Here's an example. Let's assume we have the equation 2x + 3 = 7. Identify the unknown, which is 'x', and the knowns which are the numbers 2, 3, and 7. Solve for 'x' by subtracting 3 from both sides to get 2x = 4, and then divide both sides by 2 to find x = 2. This equation has one solution, which is x=2.
In cases where we have a statement like 0x=0 after simplifying, it means the equation has infinite solutions, as any value of 'x' will satisfy it. Conversely, if we get an impossibility like 0x=5, the equation has no solutions, as no value of 'x' will make this statement true.