Solve for x: (2/5)x + 1014 + 4/5 = (5/9)x + 1019 + 4/9.

Step-by-step solution

1. 1

   Combine the constants on each side. Each side has a whole number plus a proper fraction, which merge into mixed numbers:

   $1014+\frac45=1014\tfrac45=\frac{5074}{5},\qquad 1019+\frac49=1019\tfrac49=\frac{9175}{9}.$

   Converting straight to improper fractions now means the rest of the problem is pure fraction arithmetic with no mixed-number bookkeeping.

2. 2

   Gather the $x$ terms on one side and the constants on the other. Subtracting $\tfrac25x$ and $\tfrac{9175}{9}$ from both sides:

   $\frac{5074}{5}-\frac{9175}{9}=\left(\frac59-\frac25\right)x.$

   Moving the smaller coefficient $\tfrac25$ keeps the $x$ coefficient positive, avoiding a sign flip later.

3. 3

   Compute the coefficient of $x$. Over the common denominator $45$:

   $\frac59-\frac25=\frac{25}{45}-\frac{18}{45}=\frac{7}{45}.$

   The denominators $5$ and $9$ are coprime, so $45$ is the least common denominator for the whole problem — the same $45$ will reappear on the left.

4. 4

   Compute the constant difference. Again over $45$:

   $\frac{5074}{5}-\frac{9175}{9}=\frac{5074\cdot9-9175\cdot5}{45}=\frac{45666-45875}{45}=\frac{-209}{45}.$

   Both $45666$ and $45875$ are worth checking: $5074\times9=45666$ and $9175\times5=45875$.

5. 5

   Solve and check exactly. The equation is now $\frac{-209}{45}=\frac{7}{45}x$, and multiplying both sides by $45$ makes the denominators vanish:

   $-209=7x\;\Longrightarrow\;x=-\frac{209}{7}\approx-29.857.$

   Since $209=11\times19$ this does not reduce. Substituting back with exact fractions: both sides evaluate to $\frac{7020}{7}$ ✓.