solve for x,

1.75e^-x*0.5 + 1.75e^-x + 1.75e^-x*1.5 + 101.75e^-x*2 = 86.3995

We are given the equation:

\[
1.75e^{-x} \cdot 0.5 + 1.75e^{-x} + 1.75e^{-x} \cdot 1.5 + 101.75e^{-x} \cdot 2 = 86.3995
\]

### Step 1: Factor out \( e^{-x} \)
We notice that \( e^{-x} \) is a common factor in all terms on the left-hand side. Let's factor it out:

\[
e^{-x} \cdot \left( 1.75 \cdot 0.5 + 1.75 + 1.75 \cdot 1.5 + 101.75 \cdot 2 \right) = 86.3995
\]

### Step 2: Simplify the constants inside the parentheses
Now let's compute the sum inside the parentheses:

\[
1.75 \cdot 0.5 = 0.875
\]
\[
1.75 \cdot 1.5 = 2.625
\]
\[
101.75 \cdot 2 = 203.5
\]

So, the equation becomes:

\[
e^{-x} \cdot \left( 0.875 + 1.75 + 2.625 + 203.5 \right) = 86.3995
\]

Simplifying the sum:

\[
0.875 + 1.75 + 2.625 + 203.5 = 208.75
\]

Thus, the equation reduces to:

\[
e^{-x} \cdot 208.75 = 86.3995
\]

### Step 3: Solve for \( e^{-x} \)
To isolate \( e^{-x} \), divide both sides by 208.75:

\[
e^{-x} = \frac{86.3995}{208.75}
\]

Calculating the right-hand side:

\[
e^{-x} \approx 0.4139
\]

### Step 4: Solve for \( x \)
To solve for \( x \), take the natural logarithm of both sides:

\[
-x = \ln(0.4139)
\]

\[
x = -\ln(0.4139)
\]

Now, let's compute \( \ln(0.4139) \):

\[
\ln(0.4139) \approx -0.8827
\]

Thus:

\[
x \approx 0.8827
\]

### Final Answer:
\[
x \approx 0.8827
\]

Would you like more details on any part of the solution?

Here are 5 related questions to deepen your understanding:
1. What is the general method for solving equations involving \( e^x \)?
2. How do you factor common terms in an exponential equation?
3. What is the significance of taking the natural logarithm to solve for \( x \)?
4. How can you solve a system of exponential equations?
5. What are some real-world applications of exponential decay and growth?

**Tip:** When solving exponential equations, always try to factor out common terms to simplify the equation before solving.

solve for x, 1.75e^-x*0.5 + 1.75e^-x + 1.75e^-x*1.5 + 101.75e^-x*2 = 86.3995

Discover how to solve the exponential equation 1.75e^-x*0.5 + 1.75e^-x + 1.75e^-x*1.5 + 101.75e^-x*2 = 86.3995 with a comprehensive step-by-step guide. This solution involves factoring out the common term, simplifying constants, and using logarithms to find x. Suitable for high school students in Grades 10-12.

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