Question

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. Parallel to the line \(y=5 ;\) containing the point (4,2)

Short answer

The equation of the line is \(y = 2\).

Step-by-step solution

- Understand the slope of the given line

The equation given is in the form of \[y = mx + b\] where \(m\) is the slope. For the line \(y = 5\), it can be seen that it is a horizontal line with slope \(m = 0\).

- Identify the slope of the new line

Since our new line must be parallel to \(y = 5\), it will have the same slope as that line. Hence, the slope \(m\) of our new line is also 0.

- Use the given point to find the equation

We need to find the equation of the line that passes through the point (4,2) and has a slope of 0. The general form for the slope-intercept equation is \[ y = mx + b \]. Given \(m = 0\), our equation becomes \[ y = 0x + b \]. Now, substitute the point (4,2) into the equation: \(2 = 0*4 + b\), which simplifies to \(2 = b\).

- Write the final equation

Substitute \(b = 2\) back into the slope-intercept form equation: \[y = 0x + 2\], which simplifies to \(y = 2\).

Key concepts

slope-intercept form

Understanding the slope-intercept form of a line is essential for solving various problems in algebra and geometry. The slope-intercept form is given by the equation \(y = mx + b\), where * \(m\) represents the slope of the line, which measures how steep the line is. * \(b\) represents the y-intercept, which is where the line crosses the y-axis.

This form is very useful because it directly shows the slope and the y-intercept, making graphing and analysis straightforward. If you understand how to manipulate \(m\) and \(b\), you can easily find equations of lines that match specific criteria.

parallel lines

Parallel lines are lines in a plane that do not intersect. They have the same slope but different y-intercepts. In the problem, we are asked to find a line parallel to \(y = 5\).

Since \(y = 5\) is a horizontal line, its slope, \(m\), is 0. Thus, any line parallel to it will also have a slope of 0. This is key to solving the problem because it allows us to maintain the slope while adjusting the y-intercept to match the given point (4,2). So, if you see the word 'parallel', always remember: parallel lines = same slope.

slope

The slope of a line describes its steepness and direction. It is calculated as the 'rise over the run', or the change in the y-values divided by the change in the x-values (∆y/∆x).

In mathematical terms, if you have two points, \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) is computed as \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

In the context of the problem given, since our reference line is horizontal (\(y = 5\)), the slope is 0. This simplifies our work because the new line, which must be parallel to it, will also have a slope of 0. A zero slope indicates a horizontal line. This characteristic is straightforward and significantly simplifies problems involving horizontal lines.