Let's analyze the given equation step by step:
### Given Equation:
[tex]\[ 20x + 25 = 5(2x + 21) \][/tex]
### Step 1: Expand the Right-hand Side
First, distribute the 5 on the right-hand side:
[tex]\[ 20x + 25 = 5 \cdot 2x + 5 \cdot 21 \][/tex]
[tex]\[ 20x + 25 = 10x + 105 \][/tex]
### Step 2: Rearrange to Simplify
Next, we can bring all the [tex]\(x\)[/tex] terms to one side and all the constant terms to the other side. Subtract [tex]\(10x\)[/tex] from both sides and also subtract 25 from both sides to simplify:
[tex]\[ 20x + 25 - 10x = 10x + 105 - 10x \][/tex]
[tex]\[ 10x + 25 = 105 \][/tex]
[tex]\[ 10x = 105 - 25 \][/tex]
[tex]\[ 10x = 80 \][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Now, solve for [tex]\(x\)[/tex] by dividing both sides by 10:
[tex]\[ x = \frac{80}{10} \][/tex]
[tex]\[ x = 8 \][/tex]
### Step 4: Nature of the Equation
Now, determine the nature of the equation:
To classify whether the equation is an identity, a conditional equation, or a contradiction, we check:
- An identity is an equation that holds true for all values of the variable.
- A conditional equation holds true for certain values of the variable.
- A contradiction is an equation that has no solution.
This equation holds true only for [tex]\(x = 8\)[/tex]. Thus, it is a conditional equation.
### Solution Set
The solution set for the equation [tex]\(20x + 25 = 5(2x + 21)\)[/tex] is:
[tex]\[ x = 8 \][/tex]
### Conclusion
The given equation is a conditional equation with the solution set:
[tex]\[ \{8\} \][/tex]
