Identifying the Consistent Solution for Equations An equation represents a relationship between two variables, often denoted by x and y. Solving an equation...
An equation represents a relationship between two variables, often denoted by x and y. Solving an equation means finding the values of y that make the equation true for a given value of x. However, there can be multiple solutions, depending on the specific equation and the values of the variables.
The consistent solution is a set of values for y that satisfy the equation for all possible values of x that are admissible (within the domain of the variables). These solutions represent the values of y that make the equation true and represent the solutions to the equation.
For example, consider the equation x + y = 5. This equation has two solutions:
y = 5 - x
y = -x + 5
Therefore, the consistent solution for this equation is {5, 5}. This means that the equation is satisfied by the values of x = 0 and x = 5, and it has two distinct solutions for other values of x.
Key characteristics of the consistent solution:
It is the set of all possible y-values that can be obtained from the equation for all admissible x-values.
It represents the only valid y-values that make the equation true.
It can be found by solving the equation for y in terms of x and evaluating the resulting expression for each admissible x-value.
Identifying the consistent solution can be done by:
Solving the equation for y in terms of x.
Substituting the resulting expression for y into the original equation.
Checking if the resulting equation is satisfied for all admissible x-values.
If the equation is satisfied for all x-values, then the solution set is the consistent solution