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Chapter 2: Problem 29

Find an equation of the line that passes through the point \((-2,2)\) and is parallel to the line \(2 x-4 y-8=0\).

Short Answer

Expert verified
The equation of the line that passes through the point \((-2, 2)\) and is parallel to the line \(2x - 4y - 8 = 0\) is \(\boxed{y = \frac{1}{2}x + 3}\).

Step by step solution

01

Determine the slope of the given line

To determine the slope of the line \(2x - 4y - 8 = 0\), let's first rewrite the equation in slope-intercept form. The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. Starting with the equation \(2x - 4y - 8 = 0\), isolate y as follows: \(4y = 2x - 8\) \(y = \frac{1}{2}x - 2\) Now that the equation is in slope-intercept form, we can see that the slope of the line is \(m = \frac{1}{2}\).
02

Use the point-slope form of a linear equation

Since the slope of the desired line is the same as the slope of the given line, and the new line passes through the point \((-2, 2)\), we can use the point-slope form of a linear equation to write the equation of this line. The point-slope form of a linear equation is given by: \(y - y_1 = m(x - x_1)\) where \((x_1, y_1)\) is the point that the line passes through, and \(m\) is the slope of the line.
03

Plug in the given point and slope into the point-slope form equation

Plugging in the values \(x_1 = -2\), \(y_1 = 2\), and \(m = \frac{1}{2}\), we get: \(y - 2 = \frac{1}{2}(x - (-2))\)
04

Simplify the equation

Now, simplify the equation to get the final equation of the line: \(y - 2 = \frac{1}{2}(x + 2)\) To express the equation in slope-intercept form, distribute the slope to the terms in the parentheses and solve for y: \(y - 2 = \frac{1}{2}x + 1\) \(y = \frac{1}{2}x + 3\) So, the equation of the line that passes through the point \((-2, 2)\) and is parallel to the line \(2x - 4y - 8 = 0\) is: \(\boxed{y = \frac{1}{2}x + 3}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form is a powerful, user-friendly way to express the equation of a line. You can easily spot the slope and the y-intercept by looking at the equation. Here's how it works:
\[y = mx + b\]
Here, \(m\) represents the slope of the line, which tells you how steep the line is and in which direction it tilts. The \(b\) part is called the y-intercept, showing where the line crosses the y-axis. Simply put, if you know the slope and where the line cuts across the y-axis, you can draw the entire line.
  • The slope \(m\) determines how much the line rises or falls as you move from left to right.
  • The y-intercept \(b\) provides a starting point on the graph. It's the value of \(y\) when \(x = 0\).
A line with a large positive \(m\) will be steep going upwards, while a negative \(m\) slopes downwards. If you have zero slope, your line is entirely horizontal.
Point-Slope Form
The point-slope form is essential when you know a point through which a line passes and its slope, but not its y-intercept. It allows you to create the line equation efficiently. The formula is:
\[y - y_1 = m(x - x_1)\]
In this form, \((x_1, y_1)\) is a known point on the line, and \(m\) is the slope. With these components, you can establish the entire line. This form is particularly useful in situations when:
  • You already know two points on the line and need to calculate the slope first.
  • You want to quickly create the equation of a line without converting to slope-intercept form immediately.
Once you plug in your known values, you can manipulate the equation further into slope-intercept form, if needed, to better interpret or graph it.
Parallel Lines
Parallel lines are fascinating in geometry because they share the same slope but are distinct lines on the coordinate plane. What's important about them is that they never meet or cross each other. To recognize and construct them, remember these key points:
  • Parallel lines have the exact same slope \(m\).
  • They remain equidistant from each other at all times.
  • You can use the slope of a known line to create a new line parallel to it, given a specific point the new line must pass through.
The concept of parallel lines helps with analyzing geometrical figures and solving various algebraic problems. For example, when creating a line parallel to a given line, we use the same slope but adjust the y-intercept according to the new line's crossing point on the y-axis. Understanding parallel lines provides insights that help solve real-world problems, such as determining consistent street orientations in city planning.

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