- Start with the equation \(-x + \frac{3}{7} = 2x - \frac{25}{7}\).
- Add \(x\) to both sides: \(\frac{3}{7} = 3x - \frac{25}{7}\).
- Add \(\frac{25}{7}\) to both sides to get \(\frac{28}{7} = 3x\), which simplifies to \(4 = 3x\).
- Divide by 3 to reach the result \(\boxed{x = \frac{4}{3}}\).
Explanation
- Analyzing the Equation
The objective is to find the value of \(x\) that satisfies the linear equation \(-x + \frac{3}{7} = 2x - \frac{25}{7}\). This equation involves a single variable \(x\) and constant terms expressed as fractions with a common denominator of 7. To solve for \(x\), we need to isolate the variable by grouping all terms containing \(x\) on one side and all constant terms on the other.
- Isolating the Variable Term
To isolate \(x\), we first add \(x\) to both sides of the equation:
\[-x + x + \frac{3}{7} = 2x + x - \frac{25}{7}\]
This simplifies to:
\[\frac{3}{7} = 3x - \frac{25}{7}\]
- Combining Constants and Simplifying
Next, we add \(\frac{25}{7}\) to both sides to gather the constants:
\[\frac{3}{7} + \frac{25}{7} = 3x\]
Combining the fractions on the left side, we get:
\[\frac{28}{7} = 3x\]
Since \(\frac{28}{7} = 4\), the equation simplifies to:
\[4 = 3x\]
- Solving for x
Finally, we divide both sides by 3 to solve for \(x\):
\[x = \frac{4}{3}\]
Comparing this result with the given choices, we see that it corresponds to the second option.
Examples
Linear equations are used in many real-life situations, such as calculating the cost of an item based on a fixed base price and a variable rate. For instance, if a service costs \$3 plus a variable amount that increases over time, and you compare this to another service's pricing, you can set up a linear equation to find the exact point where both services cost the same. This allows you to make informed decisions about which service is more economical for your needs.