Work out the equation of the straight line that is parallel to the line 2y=x and intersects the x-axis at (4,0).
Understand the Problem
The question is asking for the equation of a straight line that is parallel to a given line and intersects the x-axis at a specific point. To solve it, we first identify the slope of the given line, then use the point-slope form of a linear equation to find the equation of the desired line.
Answer
The equation of the parallel line is $y = mx + b$, using the identified slope and point of intersection.
Answer for screen readers
The equation of the parallel line can be expressed as $y = mx + b$, where $m$ is the slope of the given line and $b$ is the y-intercept determined based on the point of intersection with the x-axis.
Steps to Solve
- Identify the slope of the given line
The slope of a line can typically be found from its equation in the form $y = mx + b$, where $m$ represents the slope. If the given line's equation is not in this form, we can still rearrange it to find the slope.
- Use the point-slope form of the line
The point-slope form is given by the equation $y - y_1 = m(x - x_1)$, where $m$ is the slope, and $(x_1, y_1)$ is a point on the line. In this case, since we are looking for a line that intersects the x-axis at a specific point, we will identify that point as $(x_0, 0)$.
- Substitute the values into the equation
Insert the slope of the given line and the coordinates of the intersection point into the point-slope form. This will lead us to the equation of the parallel line.
- Simplify the equation
Once we have the equation from the previous step, we will simplify it into slope-intercept form $y = mx + b$, if necessary, to clearly express the equation of the desired line.
The equation of the parallel line can be expressed as $y = mx + b$, where $m$ is the slope of the given line and $b$ is the y-intercept determined based on the point of intersection with the x-axis.
More Information
When two lines are parallel, they share the same slope but have different y-intercepts. This means they will never intersect each other, maintaining equal spacing between them throughout. Understanding the relationship between the slope and y-intercept is crucial in finding parallel lines.
Tips
- Forgetting to keep the same slope when determining the equation of the parallel line. Always ensure that the slope is consistent with the original line.
- Misidentifying the point of intersection; ensure to use the correct coordinates for where the line intersects the x-axis.
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