Solve for x: $\frac{x}{7}+\frac{x}{3}=10$

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02-11-2026
Mathematics

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Solve for x: $\frac{x}{7}+\frac{x}{3}=10$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T21:05:26-05:00

Identify the denominators as 7 and 3 and find their least common multiple, which is 21.
Multiply every term by 21 to simplify the equation to $3x + 7x = 210$.
Combine like terms to get $10x = 210$.
Solve for $x$ by dividing by 10, resulting in $x = \boxed{21}$.

Explanation

Problem Analysis

We are given the linear equation $\frac{x}{7} + \frac{x}{3} = 10$. Our goal is to solve for the variable $x$. This involves fractions with different denominators, which we can eliminate by finding a common denominator or multiplying by the least common multiple (LCM) of the denominators 7 and 3.

Eliminating Fractions

To eliminate the fractions, we identify the denominators as 7 and 3. The least common multiple (LCM) of 7 and 3 is $7 \times 3 = 21$. We multiply every term in the equation by 21 to clear the denominators:
\[21 \cdot \left(\frac{x}{7} + \frac{x}{3}\right) = 21 \cdot 10\]
Distributing the 21 gives:
\[21 \cdot \frac{x}{7} + 21 \cdot \frac{x}{3} = 210\]
Simplifying the fractions:
\[3x + 7x = 210\]

Solving for x

Now we combine the like terms on the left-hand side of the equation:
\[(3 + 7)x = 210\]
\[10x = 210\]
To isolate $x$, we divide both sides of the equation by 10:
\[x = \frac{210}{10} = 21\]
Thus, the solution to the equation is $x = 21$.

Examples

In practical scenarios, finding the common denominator is essential for managing proportional resources. For example, if you are mixing two different chemical solutions that contribute fractions of the total volume, such as $\frac{x}{7}$ and $\frac{x}{3}$ of a specific container, and you know the total amount required is 10 liters, solving this linear equation helps you determine the correct total volume $x$ needed to balance your mixture accurately. This algebraic method ensures precise resource management in science, engineering, and finance.