Let's solve each equation step-by-step and determine if they are conditional, an identity, or a contradiction.

Problem 5:

$7x - 3(2x - 5) + 5 + 3x = 4x + 20$

Step 1: Expand and simplify the equation

Distribute the $-3$ across $(2x - 5)$:

$7x - 6x + 15 + 5 + 3x = 4x + 20$

Combine like terms on the left-hand side:

$(7x - 6x + 3x) + (15 + 5) = 4x + 20$

$4x + 20 = 4x + 20$

Step 2: Analyze the equation

Since both sides of the equation are identical, we have an identity. This means that the equation is true for all values of $x$.

Problem 6:

$8x - 4x - (x - 7) + 9x + 6 = 12x - 7$

Step 1: Expand and simplify the equation

Distribute the negative sign across $(x - 7)$:

$8x - 4x - x + 7 + 9x + 6 = 12x - 7$

Combine like terms on the left-hand side:

$(8x - 4x - x + 9x) + (7 + 6) = 12x - 7$

$12x + 13 = 12x - 7$

Step 2: Isolate the variable

Subtract $12x$ from both sides:

$13 = -7$

This is clearly a contradiction, as 13 does not equal -7. This means there is no solution to this equation.

Summary:

• Problem 5 is an identity.
• Problem 6 is a contradiction.

Would you like more details or have any questions?

Here are five related questions to expand this topic:

1. What is the difference between an identity and a conditional equation?
2. How can you tell if an equation is a contradiction before solving it completely?
3. What happens when the variable cancels out during equation solving?
4. Can contradictions arise from systems of equations? How do they manifest?
5. How does the process of distributing terms affect solving equations?

Tip: Always simplify both sides of an equation as much as possible before attempting to solve it. This helps identify patterns like contradictions or identities faster.