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Chapter 2: Problem 28

Use the given conditions to write an equation for each line in point slope form and slope-intercept form. Passing through \((-2,0)\) and \((0,2)\)

Short Answer

Expert verified
The equation of the line passing through points (-2,0) and (0,2), in both point-slope form and slope-intercept form, is \( y = x + 2 \).

Step by step solution

01

Calculate the slope

The slope \( m \) of a line is calculated using two points on the line \((x_1,y_1)\) and \((x_2,y_2)\) by the formula \( m = (y_2 - y_1) / (x_2 - x_1) \). Here, our points are \(-2,0\) and \((0,2)\). Therefore, \( m = (2 - 0) / (0 - (-2)) = 2/2 = 1. \)
02

Substitute into the point-slope form of a line

The point-slope form of a line is given by \( y - y_1 = m(x - x_1) \). Given that the slope \( m = 1 \), and we can use the point \(-2,0\) (or the point \((0,2)\)), the equation of the line in point-slope form becomes \(y - 0 = 1(x - (-2)),\) which simplifies to \( y = x + 2. \)
03

Substitute into the slope-intercept form of a line

Slope-intercept form of a line is \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. Given the slope \( m = 1 \) from Step 1, and \( y = x + 2, \) the equation of the line in slope-intercept form is exactly the same as in point-slope form, i.e. \( y = x + 2 \). The y-intercept \( c \) is \(2\).

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Key Concepts

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

Point-Slope Form
The point-slope form is an equation used to describe a straight line using a known point and the slope of the line. It is a handy tool when you have a specific point
  • \( (x_1, y_1) \)
  • and the slope \( m \).
The formula for this is: \( y - y_1 = m(x - x_1) \).

This form allows you to easily plug in values, making it very practical for finding an equation of the line from a graph or from known points. For example, using the point
  • \((-2, 0)\)
  • and a calculated slope of \(1\)
we utilize the equation to get\( y - 0 = 1(x - (-2)) \),
which simplifies to \( y = x + 2 \).
This step-by-step mapping from known point and slope directly to the equation makes this form straightforward and direct.
Slope-Intercept Form
The slope-intercept form is perhaps the most commonly used linear equation format because it clearly highlights the slope and the y-intercept.
This form is convenient for quickly identifying how a line behaves. The formula is:
\( y = mx + c \).
Here:
  • \( m \) represents the slope.
  • \( c \) represents the y-intercept, where the line crosses the y-axis.
To transition the point-slope form
  • \( y - 0 = 1(x + 2) \)
to slope-intercept form, solve for \( y \).
Thus, \( y = 1x + 2 \).
An example shows that the slope is \(1\) and the y-intercept is \(2\), offering a clear picture of the line's path on a graph.
Calculating Slope
Slope is a measure of how steep a line is. To find the slope of a line, we need two points on that line.
Let's say we have points
  • \((x_1, y_1)\)
  • and \((x_2, y_2)\),
the formula to calculate the slope \( m \) is:\( m = \frac{y_2 - y_1}{x_2 - x_1} \).

For example, with points
  • \((-2, 0) \)
  • and \((0, 2)\),
the slope becomes \( m = \frac{2 - 0}{0 - (-2)} = \frac{2}{2} = 1 \).
This calculation tells us the line rises by 1 unit for every 1 unit it runs, illustrating a perfect diagonal ascent from left to right.
Line Equation Formulation
Formulating the equation of a line means using known values, such as points and slope, to build a complete mathematical representation.

The whole process can be outlined as:
  • First, determine the slope \( m \) from two points
  • Next, insert the slope and one of the known points into the point-slope form \( y - y_1 = m(x - x_1) \)
  • Simplify to formulate the equation, either keeping it in point-slope form or converting to slope-intercept form \( y = mx + c \).
In the exercise using points
  • \((-2,0)\)
  • and \((0,2)\),
we calculate the slope as \( m = 1 \);
then choose a point
  • (say \((-2,0)\))
to plug into point-slope form: \( y - 0 = 1(x - (-2)) \),
resulting in the same slope-intercept form: \( y = x + 2 \).
Understanding this logical flow is essential for tackling any line-related problem in mathematics.

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Most popular questions from this chapter

In your own words, describe how to find the midpoint of a line segment if its endpoints are known.

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