Solve:
$\frac{3x+1}{5}+\frac{x-6}{4}+\frac{x}{10}=12$
The objective is to solve the linear equation for \(x\):
\[\frac{3x+1}{5} + \frac{x-6}{4} + \frac{x}{10} = 12\]
The equation contains three terms on the left-hand side with denominators 5, 4, and 10. To solve for \(x\), we first eliminate the fractions by multiplying the entire equation by the least common multiple (LCM) of these denominators. The denominators are 5, 4, and 10. Their prime factorizations are \(5=5\), \(4=2^2\), and \(10=2 \times 5\). The LCM is \(2^2 \times 5 = 20\).
Multiplying both sides of the original equation by 20 gives:
\[20 \cdot \left( \frac{3x+1}{5} + \frac{x-6}{4} + \frac{x}{10} \right) = 20 \cdot 12\]
Applying the distributive property:
\[\frac{20(3x+1)}{5} + \frac{20(x-6)}{4} + \frac{20x}{10} = 240\]
This simplifies to:
\[4(3x+1) + 5(x-6) + 2x = 240\]
Now, expand the terms:
\[12x + 4 + 5x - 30 + 2x = 240\]
Combine the like terms (the terms with \(x\) and the constant terms):
\[(12x + 5x + 2x) + (4 - 30) = 240\]
\[19x - 26 = 240\]
Now we isolate \(x\). Add 26 to both sides of the equation:
\[19x = 240 + 26\]
\[19x = 266\]
Finally, divide both sides by 19:
\[x = \frac{266}{19}\]
Dividing 266 by 19 results in:
\[x = 14\]
To verify, substitute \(x=14\) back into the original equation:
\[\frac{3(14)+1}{5} + \frac{14-6}{4} + \frac{14}{10} = \frac{43}{5} + \frac{8}{4} + \frac{14}{10} = 8.6 + 2 + 1.4 = 12\]
The solution is confirmed to be \(x = 14\).
Linear equations like this often appear in real-life budgeting or planning tasks. For example, if you are dividing a project budget into three different departments with varying percentages of the total, you might use an equation like this to find the total project value \(x\). Solving such equations allows you to balance resources correctly, ensuring all financial constraints are met efficiently.