Solve the equation 4x - 0.35(x - 1) = 278.435.

Step-by-step solution

1. 1

   Expand the bracket, watching both signs.

   $-0.35(x-1) = -0.35x + 0.35$

   so the equation becomes $4x - 0.35x + 0.35 = 278.435$.

2. 2

   Collect the x terms.

   $3.65x + 0.35 = 278.435$

   The coefficient $3.65$ is the same for every problem of this family — only the right-hand side changes.

3. 3

   Isolate the x term.

   $3.65x = 278.435 - 0.35 = 278.085$

4. 4

   Divide and reduce the fraction.

   $x = \frac{278.085}{3.65} = \frac{278085}{3650} = \frac{55617}{730} \approx 76.187671$

   Both numerator and denominator were divided by $5$; since $730 = 2 \times 5 \times 73$ and $55617$ is odd and not divisible by $5$ or $73$, this is lowest terms.

5. 5

   Check by substitution. $4(76.187671) = 304.750684$ and $0.35(75.187671) = 26.315685$; subtracting gives $278.434999 \approx 278.435$, confirming the root.

6. 6

   Compare with the sibling problem. With the right-hand side $253.586$ the root is $69.379726$. The difference in roots, $76.187671 - 69.379726 = 6.807945$, equals $\tfrac{278.435 - 253.586}{3.65} = \tfrac{24.849}{3.65}$ — exactly the slope relationship you would expect from a linear equation.