What is conditional, and identity or a contradiction ?

in terms of a linear equation.

Please can you give some examples

Assume that the box contains 12 balls:

5 red, 3 blue, and 4 yellow. As in the text, you draw one
ball, note its color, and if it is yellow replace it. If it is not yellow
you do not replace it. You then draw a second ball and note its color.

(1) What is the probability that the second ball drawn is yellow?

(2) What is the probability that the second ball drawn is red?

In the context of a linear equation, a conditional equation is one that has a solution, meaning it can be true for certain values of the variables in the equation. An identity equation is always true, regardless of the values of the variables involved. A contradiction equation is never true, no matter what values are assigned to the variables.

Here are some examples:

1. Conditional equation:
2x - 3 = 7
This equation has a solution. By solving it, we find that x = 5, so when x equals 5, the equation is true.

2. Identity equation:
3x + 4 = 3(x + 1) + 1
This equation is always true for any value of x. If you simplify both sides of the equation, you will see that they are equal.

3. Contradiction equation:
2x + 1 = 3x - 2
This equation has no solution. If you simplify it, you will find that the variables cancel out, leaving an equation that is not true for any value of x.

In the context of a linear equation, a conditional, an identity, and a contradiction are three different types of solutions that can arise when solving an equation.

1. Conditional: A conditional solution is one where the equation is true for certain values of the variables, but not true for all values. In other words, there is a specific set of values that satisfy the equation.

Example: Consider the linear equation 2x + 3 = 7. By solving this equation, we find that x = 2. So, the equation is true when x takes the value 2, but it is not true for any other value of x. Therefore, this is a conditional solution.

2. Identity: An identity is a solution that is true for all values of the variables. In other words, all values of the variables satisfy the equation.

Example: Let's examine the equation 3x - 4 = 3x - 4. This equation is true regardless of the value of x. If you simplify both sides of the equation, you will find that they are equal. Therefore, this is an identity solution.

3. Contradiction: A contradiction occurs when there is no value of the variables that satisfies the equation. In other words, the equation is never true for any value of the variables.

Example: Consider the equation 5x + 2 = -3x + 7. If we attempt to solve this equation, we get 8x = 5, which implies x = 5/8. However, when we substitute this value back into the original equation, it does not hold true. Therefore, there is no value of x that satisfies the equation, making it a contradiction.

By examining the solutions to a linear equation, we can determine whether it has a conditional, identity, or contradictory solution.