Verified Answer :
- Identify the slope of line \(s\) as \(m = -\frac{7}{10}\).
- Since line \(t\) is parallel to line \(s\), set its slope to \(m = -\frac{7}{10}\).
- Apply the point-slope form \(y - y_1 = m(x - x_1)\) with point \((6, 5)\).
- Isolate \(y\) to obtain the slope-intercept form: \(y = -\frac{7}{10}x + \frac{46}{5}\).
Final Answer: \(\boxed{y = -\frac{7}{10}x + \frac{46}{5}}\)
Explanation
- Analyzing the Properties of the Lines
Line \(s\) has an equation of \(y = -\frac{7}{10}x - 1\). This is in slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. Thus, the slope of line \(s\) is \(m = -\frac{7}{10}\). Since line \(t\) is parallel to line \(s\), it must have the same slope. Therefore, the slope of line \(t\) is also \(m = -\frac{7}{10}\). We are given that line \(t\) passes through the point \((6, 5)\).
- Using the Point-Slope Form
To find the equation of line \(t\), we use the point-slope form, which is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the given point. Substituting the slope \(m = -\frac{7}{10}\) and the point \((6, 5)\) into this form, we get:
\[y - 5 = -\frac{7}{10}(x - 6)\]
- Simplifying the Equation
Next, we distribute the slope \(- \frac{7}{10}\) on the right side of the equation:
\[y - 5 = -\frac{7}{10}x + \left(-\frac{7}{10}\right)(-6)\]
\[y - 5 = -\frac{7}{10}x + \frac{42}{10}\]
We simplify the fraction \(\frac{42}{10}\) by dividing both the numerator and the denominator by 2, which gives \(\frac{21}{5}\):
\[y - 5 = -\frac{7}{10}x + \frac{21}{5}\]
- Finalizing the Equation for Line \(t\). Conclusion and Final Answer.
Finally, we isolate \(y\) to write the equation in slope-intercept form by adding 5 to both sides of the equation:
\[y = -\frac{7}{10}x + \frac{21}{5} + 5\]
To add \(\frac{21}{5} + 5\), we find a common denominator:
\[\frac{21}{5} + \frac{5 \times 5}{5} = \frac{21}{5} + \frac{25}{5} = \frac{46}{5}\]
Thus, the equation for line \(t\) is:
\[y = -\frac{7}{10}x + \frac{46}{5}\]
This equation is in the requested slope-intercept form with simplified fractions.
Examples
In urban planning and architecture, determining parallel lines is essential when designing roads or structural beams that must maintain a constant distance from each other. For instance, if you are planning two parallel bike lanes on a grid where one follows a specific street layout, the mathematical determination of the second line ensures consistent alignment. This application of linear equations allows planners to calculate precise coordinates for infrastructure, ensuring that new additions maintain the desired geometric relationship with existing features.