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Chapter 4: Problem 44

Find an equation of the line containing the two given points. Express your answer in the indicated form. \((-3.4,5.8)\) and \((-1.8,3.4) ;\) slope-intercept form

Short Answer

Expert verified
The equation of the line containing the given points in slope-intercept form is \(y = -1.5x + 0.7\).

Step by step solution

01

Calculate the slope

Recall the formula for calculating the slope (m) between two points (x1, y1) and (x2, y2): \[m = \frac{y2-y1}{x2-x1}\] Using the given points \((-3.4,5.8)\) and \((-1.8,3.4)\), the slope can be calculated as: \[m = \frac{3.4 - 5.8}{-1.8 - (-3.4)}\]
02

Use the point-slope form

Using the point-slope form, the equation of the line passing through the point \((-3.4,5.8)\) and having the slope \(m\) can be written as: \(y - 5.8 = m(x +3.4)\) Now, we need to find the value of 'm' from step 1 and substitute it in the equation. From step 1, \[m = \frac{3.4 - 5.8}{-1.8 - (-3.4)} = \frac{-2.4}{1.6} = -1.5\] Substitute m in the equation: \(y - 5.8 = -1.5(x +3.4)\)
03

Convert to slope-intercept form

To convert the equation to slope-intercept form, we need to simplify the equation and isolate 'y'. \(y - 5.8 = -1.5(x +3.4)\) Distribute the -1.5 into the parentheses: \(y - 5.8 = -1.5x - 5.1\) Add 5.8 to both sides of the equation: \(y = -1.5x + 0.7\) The equation of the line containing the given points in slope-intercept form is \(y = -1.5x + 0.7\).

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Key Concepts

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

Calculating the Slope
The slope of a line is a measure of its steepness and direction. When given two points on the line, calculating the slope is straightforward. You use the formula:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
In this formula, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. In our exercise, we have the points \((-3.4, 5.8)\) and \((-1.8, 3.4)\). To find the slope (\(m\)), you subtract the \(y\)-coordinates and \(x\)-coordinates of these two points:
  • First, calculate \(3.4 - 5.8\) to find the difference in \(y\)-coordinates.
  • Then, calculate \(-1.8 - (-3.4)\) to find the difference in \(x\)-coordinates, which is the same as \(-1.8 + 3.4\).
This gives us a slope \(m = \frac{-2.4}{1.6} = -1.5\). The negative sign means the line slopes downwards as you move from left to right.
Equation of a Line
The equation of a line in a coordinate plane can be expressed in various forms. The most common forms include:
  • Slope-intercept form: \(y = mx + b\)
  • Point-slope form: \(y - y_1 = m(x - x_1)\)
  • Standard form: \(Ax + By = C\)
In this exercise, we're asked to find the equation in slope-intercept form. This form clearly shows the slope \(m\), and the \(y\)-intercept \(b\) on the graph. Once we know the slope and a point on the line, we can translate these into the equation. If you need to convert from another form to slope-intercept form, you simply solve for \(y\) to isolate it on one side of the equation. Remember that when rearranging, each arithmetic step keeps the line equation equivalent, just represented differently.
Point-Slope Form
The point-slope form of a line's equation is quite useful when you have a point on the line and know the slope. It's expressed as:
  • \(y - y_1 = m(x - x_1)\)
The \(m\) in the equation represents the slope, while \((x_1, y_1)\) represents a specific point on the line. In our problem, since we have the point \((-3.4, 5.8)\) and the slope \(-1.5\), we plug these into the point-slope formula:
  • \(y - 5.8 = -1.5(x + 3.4)\)
This equation represents the same line and shows how each change in \(x\) affects \(y\) based on the slope. From this form, you can easily convert to slope-intercept form by distributing the slope and making \(y\) the subject.
Coordinate Geometry
Coordinate geometry, or analytic geometry, is all about using coordinates to represent geometric shapes and their relationships. In this exercise, we're dealing with a line, one of the simplest geometric shapes. By using coordinate geometry, we understand important properties such as:
  • Slope and its implication on line orientation.
  • Intercepts which give an understanding of where the line crosses the axes.
  • Distance between points if needed, although not in this direct exercise.
Using coordinates, we express the relationship between variables \(x\) and \(y\). When given coordinates, we introduce a mathematical relationship that defines how \(x\) and \(y\) change relative to each other, resulting in an equation of a line. Coordinate geometry helps in visualizing and interpreting how algebraic equations manifest as geometrical lines, curves, and shapes on a graph.

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Most popular questions from this chapter

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(m=-1\) and contains \((8,-8)\)

Graph each function by finding the \(x\) - and \(y\) -intercepts and one other point. $$f(x)=-x$$

Write an equation of the line perpendicular to the given line and containing the given point. Write the answer in slope-intercept form or in standard form, as indicated. \(x+y=9 ;(-5,-5) ;\) slope-intercept form

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{1}{x+10}$$

Graph each function using the slope and \(y\) -intercept. $$g(x)=\frac{1}{2} x-\frac{3}{2}$$
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