What is the Equation of a Straight Line Passing Through the Point (0, 7) and Parallel to the Line of the Equation y 2x - 5?
In this article, we'll explore the concept of parallel lines, focusing on finding the equation of a line that is parallel to y 2x - 5 and passes through the point (0, 7). We'll walk through the step-by-step process and explain the related mathematical concepts.
Understanding the Slope of a Line
First, let's examine the given line y 2x - 5. To determine if two lines are parallel, we need to know the slope of each line. A line's slope is a measure of its steepness and is represented by the letter m. The equation y mx b is the slope-intercept form, where m is the slope and b is the y-intercept.
The given line y 2x - 5 is already in slope-intercept form, allowing us to identify its slope directly. The slope, m_1, of the given line is 2.
Determining the Slope of the Parallel Line
Two lines are parallel if and only if they have the same slope. Therefore, the slope of the line parallel to the given line is also 2. This is because parallel lines maintain a constant distance apart and have identical angles of inclination.
Using the Point-Slope Form to Find the Equation of the Line
To find the equation of the line that is parallel to y 2x - 5 and passes through the point (0, 7), we can use the point-slope form of a line. The point-slope form of a line is given by:
y - y_1 m(x - x_1), where (x_1, y_1) is a point on the line and m is the slope of the line.
In this case, we know the slope m 2 and the point (x_1, y_1) (0, 7). Substituting these values into the point-slope form, we get:
y - 7 2(x - 0)
Now, let's simplify this equation:
y - 7 2x
Adding 7 to both sides of the equation:
y 2x 7
This is the equation of the line that is parallel to y 2x - 5 and passes through the point (0, 7).
Standard Form of the Line Equation
The equation in the standard form, Ax By C, can also be derived from the slope-intercept form. The equation y 2x 7 can be rewritten as:
2x - y 7 0
Which is the standard form of the equation.
Summary and Conclusion
In summary, the equation of a line parallel to y 2x - 5 and passing through the point (0, 7) is y 2x 7. This result was obtained by using the fact that parallel lines have the same slope and applying the point-slope form of a line equation.