Solve for x: 6.58x + 14 = 6.75x.

Step-by-step solution

1. 1

   Recognise the shape. Two quantities grow at almost the same rate — $6.58$ and $6.75$ per unit — but one starts $14$ ahead. This is a break-even equation: it asks when the faster one catches up. Because the rates are close, expect a large answer.

2. 2

   Collect the x terms on the side with the larger coefficient. Subtracting $6.58x$ from both sides keeps the remaining coefficient positive:

   $14=6.75x-6.58x$

3. 3

   Subtract the decimals carefully. Aligning the hundredths:

   $6.75-6.58=0.17\qquad\Longrightarrow\qquad 14=0.17x$

   That tiny $0.17$ is the whole gap in growth rate, and it is what makes the answer large.

4. 4

   Divide by 0.17.

   $x=\frac{14}{0.17}=\frac{1400}{17}$

   Multiplying numerator and denominator by $100$ clears the decimal, and $17$ is prime so the fraction is already reduced.

5. 5

   Evaluate the decimal.

   $x=\frac{1400}{17}=82.352941\ldots\approx82.35$

   As a rough check, $0.17\times82=13.94$, just under $14$ ✓.

6. 6

   Verify in the original equation. Left: $6.58\times82.352941+14=541.882353+14=555.882353$. Right: $6.75\times82.352941=555.882353$ ✓. Both sides agree to six decimals.