To find the equation of a line that is parallel to another line and passes through a specific point, follow these steps:
Step 1: Identify the slope of the given line.
The given line is \( y = x + 4 \). In slope-intercept form, which is \( y = mx + b \), ‘m’ represents the slope of the line, and ‘b’ represents the y-intercept. Here, the coefficient of \( x \) is \( 1 \), therefore the slope \( m \) of the given line is \( 1 \).
Step 2: Use the same slope for the new line.
Since parallel lines have the same slope, the slope of the line we are looking for will also be \( 1 \).
Step 3: Use the point-slope form of the equation of a line to find the equation of the new line.
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is the point through which the line passes and \( m \) is the slope of the line.
Step 4: Plug in the slope and the point into the point-slope formula.
In our case, \( m = 1 \) and the point is \( (-1, 2) \). Plugging these values into the point-slope formula, we get:
\[ y - 2 = 1(x - (-1)) \]
\[ y - 2 = 1(x + 1) \]
Step 5: Simplify the equation to slope-intercept form (optional).
To find the equation in slope-intercept form, we need to solve for \( y \):
\[ y - 2 = x + 1 \]
Add \( 2 \) to both sides:
\[ y = x + 1 + 2 \]
\[ y = x + 3 \]
Hence, the equation of the line that passes through the point \( (-1, 2) \) and is parallel to the line \( y = x + 4 \) is \( y = x + 3 \).
