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Chapter 4: Problem 466

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(y=5,\) point (2,-2)

Short Answer

Expert verified
The equation is \(y = -2\).

Step by step solution

01

- Understand the given line

The given line is described by the equation \(y = 5\). This is a horizontal line where every point on the line has a y-coordinate of 5.
02

- Determine the slope of the given line

A horizontal line has a slope of 0 because it does not rise or fall. Therefore, the slope (m) of the given line is 0.
03

- Use the slope for the new line

Since parallel lines have the same slope, the slope of the line we need to find is also 0.
04

- Write the equation of the new line

The general form of the slope-intercept equation is \(y = mx + b\). With a slope of 0, this simplifies to \(y = b\).
05

- Determine the y-intercept

The new line must pass through the point (2, -2). Because the slope is 0, the y-value of all points on the line is the same. Thus, the y-coordinate of the given point is the y-intercept. Therefore, \(b = -2\).
06

- Write the final equation

Substitute the y-intercept into the equation \(y = b\). The final equation of the line is \(y = -2\).

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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 way of writing the equation of a line so it's easy to understand its slope and y-intercept. The formula is given by: \[y = mx + b\] Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept, which is the point where the line crosses the y-axis. To find the equation of a line in slope-intercept form, you need to know the slope and the y-intercept:
  • If you have a point (x, y) and the slope, you can solve for \(b\).
  • Insert the slope and y-intercept values back into the formula to find the equation.
In this exercise, once we knew the slope was 0, we simplified the equation to \(y = b\), since any number multiplied by 0 remains 0.
horizontal lines
Horizontal lines are unique in that they have a slope of 0. This makes them easy to identify and work with in algebra. Every point along a horizontal line has the same y-coordinate:

In this exercise, the given line is described by the equation \(y = 5\). This tells us that no matter what value x takes, \(y\) will always be 5.

Key characteristics of horizontal lines include:
  • They are perfectly flat.
  • Their slopes are always zero.
  • The equation of a horizontal line can be written as \(y = k\), where \(k\) is the constant y-value for all points on the line.
This exercise asked us to find a line that is parallel to \(y = 5\) and passes through the point (2, -2).
equations of lines
Finding the equation of a line involves understanding its behavior and key properties:

  • Slope: Measures the steepness of the line and is represented by \(m\).
  • Y-intercept: The point where the line crosses the y-axis, represented by \(b\).
  • Parallel lines: They share the same slope but different y-intercepts.
In our exercise, we were tasked with finding an equation for a line parallel to \(y = 5\) that passes through (2, -2).

Since the lines are parallel, their slopes are identical (both 0). Using the y-coordinate of the given point, we identified the y-intercept of the new line.

So, the final equation of the new line in slope-intercept form is \(y = -2\). This method ensures we correctly find the parallel line's equation that includes the given point.

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Most popular questions from this chapter

Graph the linear inequality \(y<4\)

Find the equation of each line. Write the equation in slope-intercept form. Perpendicular to the line \(4 x+3 y=1\), containing point (0,0)

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. line \(2 x+5 y=6,\) point(0,0)

Graph the linear inequality \(y \geq 2\)

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. line \(y=\frac{2}{3} x-4,\) point (2,-4)
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