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Answer
The equation of the line parallel to \( y = x - 4 \) that passes through the point \((1, -1)\) is \( y = x - 2 \).
Solution
To find the equation of a line that passes through the point \((1, -1)\) and is parallel to the line \(y = x - 4\), follow these steps:
1. **Determine the Slope of the Given Line:**
The given line is \(y = x - 4\). This is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
- **Slope (\(m\))** of the given line: \(1\)
2. **Use the Point-Slope Form for the New Line:**
Since parallel lines have the same slope, the new line will also have a slope of \(1\).
The **point-slope form** is:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.
Plugging in the values:
\[
y - (-1) = 1(x - 1)
\]
Simplify:
\[
y + 1 = x - 1
\]
\[
y = x - 2
\]
3. **Final Equation:**
The equation of the line parallel to \(y = x - 4\) and passing through \((1, -1)\) is:
\[
y = x - 2
\]
**Graphical Representation:**
- **Original Line:** \(y = x - 4\) (Slope = 1)
- **New Line:** \(y = x - 2\) (Slope = 1, passes through (1, -1))
Both lines have the same slope, confirming they are parallel.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To find the equation of a line that passes through the point \( (1, -1) \) and is parallel to the line given by the equation \( y = x - 4 \), note that parallel lines have the same slope. The slope of \( y = x - 4 \) is \( 1 \). Using the point-slope form of a line \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point \( (1, -1) \), we can plug in the values: \[ y - (-1) = 1(x - 1) \] Simplifying this gives: \[ y + 1 = x - 1 \] Thus, the equation becomes: \[ y = x - 2 \] Now you have a line parallel to \( y = x - 4 \) that goes through the point \( (1, -1) \). Enjoy your mathematical journey!
