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Chapter 1: Problem 36

Find an equation for the line that is described. Write the answer in the two forms \(y=m x+b\) and \(A x+B y+C=0\). Is perpendicular to \(x-y+2=0\) and passes through (3,1).

Short Answer

Expert verified
The equations are \(y = -x + 4\) and \(x + y - 4 = 0\).

Step by step solution

01

Identify the Slope of the Given Line

The equation provided is \(x - y + 2 = 0\). This can be rewritten in the form \(y = mx + b\) to better identify the slope. Rearrange the equation:\[-y = -x - 2\]\[y = x + 2\]Thus, the slope \(m\) of the given line is \(1\).
02

Determine the Perpendicular Slope

A line that is perpendicular to another has a slope that is the negative reciprocal of the original line's slope. The slope of the given line is \(1\). The negative reciprocal is:\[m_{ ext{perpendicular}} = -\frac{1}{1} = -1\]
03

Use Point-Slope Form to Find the Line Equation

We need a line that has a slope of \(-1\) and passes through the point \((3,1)\). Using the point-slope form \(y - y_1 = m(x - x_1)\) where \((x_1, y_1) = (3, 1)\) and \(m = -1\):\[y - 1 = -1(x - 3)\]Simplify the equation:\[y - 1 = -x + 3\]\[y = -x + 4\]
04

Write the Equation in Two Forms

The equation \( y = -x + 4 \) is already in the slope-intercept form \( y = mx + b \). To convert it to the standard form \(Ax + By + C = 0\), rearrange the terms:\[x + y - 4 = 0\]Thus, the equation in standard form is \(x + y - 4 = 0\).

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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 of an equation of a line is one of the simplest and most common ways to represent lines in algebra. This form is written as \(y = mx + b\). Here, \(m\) represents the slope of the line, while \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
This form makes it very easy to quickly determine both the direction and steepness of a line, which is why it is so popular.
  • If \(m > 0\), the line slopes upwards.
  • If \(m < 0\), the line slopes downwards.
  • If \(m = 0\), the line is horizontal.
Using the slope-intercept form can help you graph lines quickly and understand their behavior by assessing the relationship between \(x\) and \(y\). It's a great tool for visual learning.
Standard form
The equation of a line in standard form is expressed as \(Ax + By + C = 0\). In this form, \(A\), \(B\), and \(C\) are constants, and \(A\) and \(B\) cannot both be zero. This form provides a different perspective than the slope-intercept form by focusing on anchor points in the coordinate plane.
While it may not immediately reveal the slope or y-intercept, converting from standard form to slope-intercept form can make these quantities apparent.
  • Rewriting in slope-intercept form helps find the slope \((m = -A/B)\).
  • This form is especially useful for solving systems of equations using elimination methods.
  • Easy to verify if a particular point \((x, y)\) lies on the line by substituting \(x\) and \(y\) into the equation.
The standard form is versatile and facilitates direct computations like finding perpendicular or parallel lines.
Perpendicular lines
Perpendicular lines intersect at a right angle (90 degrees). A key characteristic of perpendicular lines is their slopes, which relate to each other as negative reciprocals. If a line has a slope \(m\), the slope of a line perpendicular to it is \(-1/m\).
  • For example, if one line's slope is \(2\), a perpendicular line's slope would be \(-1/2\).
  • This concept is critical for determining orthogonal intersections or geometric figures' attributes.
  • In coordinate geometry, this property helps construct squares, rectangles, and right triangles efficiently.
Understanding perpendicular lines offers conceptual insight into the coordinate plane, enhancing geometric and algebraic problem-solving capabilities.
Point-slope form
The point-slope form is useful for writing equations of lines when you have a specific point and the slope. This form is written as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a given point on the line and \(m\) is the slope. Point-slope form is particularly handy when working through problems involving coordinates
  • This format instantly aligns with given data without requiring the y-intercept.
  • By inputting known values, it readily produces an equation of the line.
  • It facilitates easy conversion to other forms such as slope-intercept or standard form.
This approach provides flexibility and allows the direct formulation of a line, thus being invaluable in both elementary applications and intricate geometric constructions.

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Most popular questions from this chapter

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x|>1$$

In parts (a) and (b), sketch the interval or intervals corresponding to the given inequality: (a) \(|x-2|<1\) (b) \(0<|x-2|<1\) (c) In what way do your answers in (a) and (b) differ? (The distinction is important in the study of limits in calculus.)

Find an equation for the line passing through the two given points. Write your answer in the form \(y=m x+b\). (a) (7,9) and (-11,9) (b) \((5 / 4,2)\) and \((3 / 4,3)\) (c) (12,13) and (13,12)

Graph the equation after determining the \(x\) - and \(y\) -intercepts and whether the graph possesses any of the three types of symmetry described on page 58 $$y=-1 / x^{3}$$

Specify the center and radius of each circle. Also, determine whether the given point lies on the circle. $$(x+4)^{2}+(y+2)^{2}=20 ;(0,1)$$
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