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Chapter 4: Problem 39

Find an equation of the line containing the two given points. Express your answer in the indicated form. \((-1,-5)\) and \((3,-2) ;\) standard form

Short Answer

Expert verified
The equation of the line containing the given points in standard form is \(3x-4y=-17\).

Step by step solution

01

Find the slope (m) of the line

To find the slope (m) of the line, we can use the formula: \(m=\frac{y_2-y_1}{x_2-x_1}\). Plugging in the given points, we get: \[m = \frac{(-2)-(-5)}{(3)-(-1)}\]
02

Simplify the slope

Simplifying the expression for the slope, we have: \[m = \frac{3}{4}\]
03

Find the equation in point-slope form

Using the point-slope form of a linear equation, which is \(y-y_1=m(x-x_1)\), we can plug in either point and the slope. Using point \((-1, -5)\), we get: \[y - (-5) = \frac{3}{4}(x - (-1))\]
04

Simplify the equation

Simplifying the equation, we obtain: \[y + 5 = \frac{3}{4}(x + 1)\]
05

Convert to standard form

To convert our equation into standard form, we first need to eliminate the fraction by multiplying both sides of the equation by 4: \[4(y+5)=4\left(\frac{3}{4}(x+1)\right)\] This gives us the equation: \[4y+20=3(x+1)\]
06

Distribute and Simplify

Distribute and simplify the equation: \[4y+20=3x+3\] Now, we will move all the terms containing variables to the left side and the constant terms to the right side to obtain the following standard form: \[3x-4y=-17\] Thus, the equation of the line containing the given points in standard form is \(3x-4y=-17\).

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Key Concepts

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

Slope Formula
The slope formula is a crucial concept when dealing with linear equations. It is used to determine the gradient or steepness of a line that passes through two points. This formula is expressed as: \[m=\frac{y_2-y_1}{x_2-x_1}\] Here,
  • \((x_1, y_1)\) are the coordinates of the first point, and
  • \((x_2, y_2)\) are the coordinates of the second point.
The slope \(m\) tells us how much the line rises or falls as we move from one point to another. If the slope is positive, the line rises as it moves from left to right. If negative, it falls. When the slope is zero, the line is perfectly horizontal, indicating no rise or fall. In our example, the slope of the line between the points \((-1, -5)\) and \((3, -2)\) was calculated as \(\frac{3}{4}\), showing an upward trend.
Linear Equation
A linear equation represents a straight line on a coordinate plane. It is often written in the form \(y = mx + b\), where
  • \(m\) is the slope of the line, and
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
This equation provides a simple relationship between \(x\) and \(y\): for every change in \(x\), \(y\) changes by a constant amount determined by the slope, \(m\). In our task of finding the equation from given points, we initially placed the equation in point-slope form to easily transition into other forms later. By understanding how each form relates, we can solve various problems involving linear equations.
Standard Form
The standard form of a linear equation is \(Ax + By = C\), where
  • \(A\), \(B\), and \(C\) are integers, and
  • \(A\) should ideally be a positive integer.
This form is particularly useful in many fields like economics and physics since it neatly separates the input (\(x\)) from the output (\(y\)), which can help in finding intercepts and understanding relationships between variables. Given an equation in any form, converting it to standard form requires rearranging the terms: bringing \(x\) and \(y\) terms to one side and constants to the opposite. Our example, initially in point-slope form, was eventually rearranged to its standard form, \(3x - 4y = -17\), through such a process of simplification and reorganization.
Point-Slope Form
The point-slope form of an equation, given by \(y - y_1 = m(x - x_1)\), is a convenient method for writing the equation of a line when you know:
  • a point \((x_1, y_1)\) on the line
  • and the slope \(m\).
This format is particularly helpful because it directly uses the slope and a specific point. It acts as a bridge between finding a slope and converting the equation into other forms, like slope-intercept or standard form. In our solution, starting from the point \((-1, -5)\) and slope \(\frac{3}{4}\), we formed the equation: \(y + 5 = \frac{3}{4}(x + 1)\). We later transformed it into standard form, demonstrating how the point-slope form can provide an efficient starting point for various algebraic manipulations.

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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 \(y\) -intercept \((0,0)\)

Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. perpendicular to \(y=3\) containing \((2,1)\)

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{6}{5 x-3}$$

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. $$2 x+5 y=11 ;(4,2) ; \text { standard form }$$

If gasoline costs \(\$ 2.50\) per gallon, then the cost, \(C\) (in dollars), of filling a gas tank with \(g\) gal of gas is defined by $$ C(g)=2.50 g $$ a) Find \(C(8),\) and explain what this means in the context of the problem. b) Find \(C(15),\) and explain what this means in the context of the problem. c) Find \(g\) so that \(C(g)=30,\) and explain what this means in the context of the problem.
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