$\frac{10 x-3}{3}+\frac{5 x+2}{4}=5$
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$x=$
$\frac{w-4}{3}+\frac{w+1}{6}=\frac{7}{2}$
To solve the linear equations \(\frac{10x-3}{3} + \frac{5x+2}{4} = 5\) and \(\frac{w-4}{3} + \frac{w+1}{6} = \frac{7}{2}\), we must clear the denominators to isolate the variables \(x\) and \(w\). The most efficient way to do this is to multiply both sides of each equation by the least common multiple (LCM) of the denominators present in the equation. Once the denominators are removed, we will be left with standard linear equations which can be solved through algebraic rearrangement and simplification.
Consider the equation:
\[ \frac{10x-3}{3} + \frac{5x+2}{4} = 5 \]
The denominators are \(3\) and \(4\). Their least common multiple is \(12\). We multiply the entire equation by \(12\):
\[ 12 \left( \frac{10x-3}{3} + \frac{5x+2}{4} \right) = 12(5) \]
Distributing the \(12\) across the terms:
\[ 4(10x-3) + 3(5x+2) = 60 \]
\[ 40x - 12 + 15x + 6 = 60 \]
Combine like terms:
\[ 55x - 6 = 60 \]
Add \(6\) to both sides:
\[ 55x = 66 \]
Divide by \(55\):
\[ x = \frac{66}{55} = \frac{6}{5} \]
Thus, \(x = 1.2\).
Consider the equation:
\[ \frac{w-4}{3} + \frac{w+1}{6} = \frac{7}{2} \]
The denominators are \(3\), \(6\), and \(2\). Their least common multiple is \(6\). We multiply the entire equation by \(6\):
\[ 6 \left( \frac{w-4}{3} + \frac{w+1}{6} \right) = 6 \left( \frac{7}{2} \right) \]
Distributing the \(6\) across the terms:
\[ 2(w-4) + 1(w+1) = 3(7) \]
\[ 2w - 8 + w + 1 = 21 \]
Combine like terms:
\[ 3w - 7 = 21 \]
Add \(7\) to both sides:
\[ 3w = 28 \]
Divide by \(3\):
\[ w = \frac{28}{3} \]
Thus, \(w = 9\frac{1}{3}\).
We have successfully determined the values for the variables in both equations. For the first equation, the solution is \(x = \frac{6}{5}\) or \(1.2\). For the second equation, the solution is \(w = \frac{28}{3}\) or \(9\frac{1}{3}\). These steps were validated by substituting the results back into the original equations to confirm they hold true.
The final solutions are:
\(x = \frac{6}{5}\)
\(w = \frac{28}{3}\)
Linear equations are used everywhere in life, from simple financial calculations to scientific modeling. For example, if you are determining how long it takes to save for an item, you might set up an equation involving your weekly savings (\(x\)) plus an initial amount, set equal to the target price. Similarly, engineers use these equations to balance chemical mixtures, ensuring the proportions of ingredients (represented by fractions) sum to the desired total volume or concentration. Mastering this skill allows for precise planning and problem-solving in everyday tasks.