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Chapter 4: Problem 40

Find an equation of the line containing the two given points. Express your answer in the indicated form. \((-3,0)\) and \((-5,1) ;\) standard form

Short Answer

Expert verified
The equation of the line containing the two given points \((-3,0)\) and \((-5,1)\) in standard form is: \(x + 2y + 3 = 0\)

Step by step solution

01

Find the slope (m)

To find the slope of the line, we'll use the following formula: m = \(\frac{(y2 - y1)}{(x2 - x1)}\) Where (x1, y1) is the first point (-3,0) and (x2, y2) is the second point (-5, 1). Plug in the coordinates of the points into the formula. m = \(\frac{(1-0)}{(-5-(-3))}\)
02

Simplify the slope expression

Now, we'll simplify the expression to find the slope: m = \(\frac{1}{-5+3}\) m = \(\frac{1}{-2}\) So the slope of the line is -1/2.
03

Using the slope-intercept form

Now that we have the slope, we can use the slope-intercept form (y = mx + b) to find the y-intercept (b). Plug in the slope and the coordinates of one of the points (let's use (-3,0)) into the equation: 0 = \(-\frac{1}{2}\) * (-3) + b
04

Solve for the y-intercept (b)

Next, we'll solve for b: 0 = \(\frac{3}{2}\) + b b = -\(\frac{3}{2}\) So the y-intercept is -3/2.
05

Write the equation in slope-intercept form

Now that we have both the slope and y-intercept, we can write the equation of the line in slope-intercept form: y = \(-\frac{1}{2}\)x - \(\frac{3}{2}\)
06

Convert to standard form

Finally, we'll convert the equation to standard form (Ax + By = C). To do this, we'll multiply both sides by 2 (to eliminate the fractions) and then move all terms to the left side: 2y = -x - 3 x + 2y + 3 = 0 The equation of the line containing the two given points (-3,0) and (-5,1) in standard form is: x + 2y + 3 = 0

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form of a line is key to writing the equation of a line. The slope-intercept form is an equation written as \( y = mx + b \). Here:
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
This form is great for quickly identifying the slope and y-intercept directly from the equation.
For example, when you have an equation \( y = -\frac{1}{2}x - \frac{3}{2} \) from our exercise, it shows that:
  • The slope \( m = -\frac{1}{2} \)
  • The y-intercept \( b = -\frac{3}{2} \)
This form is useful because it provides information at a glance, making it easy to graph the line and understand its behavior.
Standard Form
Another way to express the equation of a line is standard form, which is written as \( Ax + By = C \). In this form:
  • \( A \), \( B \), and \( C \) are integers.
  • The equation describes the same line as when using the slope-intercept form.
An important feature of standard form is that it does not display the slope directly, unlike the slope-intercept form. However, it's well-suited for solving systems of equations as it stands equation ready.
In our given exercise, the conversion from the slope-intercept to standard form resulted in:
  • \( x + 2y + 3 = 0 \)
This showcases how you can rearrange and transform the equation to focus on particular algebraic manipulations, which can be very useful in diverse mathematical or real-world problems.
Finding Slope
The slope of a line is a measure of its steepness and indicates the direction of the line. To find the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), you use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
  • This formula subtracts the y-values and divides by the subtraction of x-values, indicating the rate at which y changes with respect to x.
In our exercise, the given points are \((-3,0)\) and \((-5,1)\). Plugging these into the formula gives:
  • \( m = \frac{1 - 0}{-5 + 3} = \frac{1}{-2} \)
This tells us that the line falls as it moves from left to right, specifically sloping down at a rate of \(-1/2\).
Understanding how to find the slope is essential because it determines how the line will appear graphically and informs whether it will rise or fall.

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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=\frac{8}{3} \text { and } y \text { -intercept }(0,-9)

Graph each function. $$k(d)=d-1$$

Write an equation of the line parallel to the given line and containing the given point. Write the answer in slope intercept form or in standard form, as indicated. $$x+4 y=32 ;(-8,5) ; \text { standard form }$$

Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. parallel to \(6 x+y=4\) containing \((-2,0)\)

Write the slope-intercept form of the equation of the line, if possible, given the following information. contains \((0,2)\) and \((6,0)\)
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