- Simplify the given equation \(2y - 2 = -2(5 - x)\) to \(y = x - 4\) to identify the slope \(m = 1\).
- Use the point-slope form \(y - y_1 = m(x - x_1)\) with slope \(m = 1\) and point \((-15, 14)\).
- Solve for \(y\) to obtain the line in slope-intercept form.
- The resulting line is \(\boxed{y = x + 29}\).
Explanation
- Finding the Slope of the Line
To find the equation of the line parallel to the given line \(2y - 2 = -2(5 - x)\) that passes through the point \((-15, 14)\), we first determine the slope of the given line. The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope.
Starting with the given equation:
\[2y - 2 = -2(5 - x)\]
We expand the right side:
\[2y - 2 = -10 + 2x\]
Adding 2 to both sides gives:
\[2y = 2x - 8\]
Dividing by 2, we obtain:
\[y = x - 4\]
The slope of this line is \(m = 1\). Parallel lines have the same slope, so the line we are looking for must also have a slope of \(m = 1\).
- Determining the Equation of the Line
We know the slope of our target line is \(m = 1\) and it must pass through the point \((-15, 14)\). Using the point-slope formula \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (-15, 14)\):
\[y - 14 = 1(x - (-15))\]
Simplifying this equation:
\[y - 14 = x + 15\]
Now, solve for \(y\) to put it in slope-intercept form:
\[y = x + 15 + 14\]
\[y = x + 29\]
The equation of the line in slope-intercept form is \(y = x + 29\).
Examples
Understanding the slopes of parallel lines is crucial in fields like civil engineering and architecture. For example, if two roads or two beams in a building are designed to be parallel, their slopes must match perfectly to maintain equal distance throughout. By calculating the equation of a parallel line, engineers can precisely position parallel structures relative to coordinate systems.