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Chapter 1: Problem 34

In Problems \(29-34\), find an equation for each line. Then write your answer in the form \(A x+B y+C=0\). Through \((4,1)\) and \((8,2)\)

Short Answer

Expert verified
Equation in the form is \(x - 4y = 0\).

Step by step solution

01

Find the Slope

Calculate the slope \(m\) of the line using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For the points \((4, 1)\) and \((8, 2)\), this becomes: \[ m = \frac{2 - 1}{8 - 4} = \frac{1}{4} \].
02

Use Point-Slope Form

Use the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), with point \((4,1)\) and slope \(\frac{1}{4}\). This gives: \[ y - 1 = \frac{1}{4}(x - 4) \].
03

Simplify to Slope-Intercept Form

Distribute the slope and simplify the equation to the slope-intercept form \(y = mx + b\): \[ y - 1 = \frac{1}{4}x - 1 \]. Add 1 to both sides to get: \[ y = \frac{1}{4}x \].
04

Write in Standard Form

Convert the equation \(y = \frac{1}{4}x\) to the standard form \(Ax + By + C = 0\). Multiply every term by 4 to eliminate the fraction: \[ 4y = x \]. Subtract \(x\) from both sides: \[ -x + 4y = 0 \]. This can also be written as \(x - 4y = 0\).

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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 of a line's equation is a very handy tool. It allows you to write an equation if you know the slope and one point on the line. The general formula looks like this: \( y - y_1 = m(x - x_1) \). Here, \( m \) is the slope, and \((x_1, y_1)\) is the point you know.
In our example, we have a slope \( m = \frac{1}{4} \) and the point \((4, 1)\). By substituting, we get: \( y - 1 = \frac{1}{4}(x - 4) \). This equation expresses the relationship between \(x\) and \(y\) for all points on the line. Understanding this formula:
  • Start with the slope: This tells you how steep the line is. It affects how much \(y\) changes when \(x\) changes.
  • Select a point: This fixes one specific location through which the line passes.
This is often the starting point for finding other forms of line equations.
Slope-intercept form
The slope-intercept form provides a clear view of a line's slope and where it crosses the y-axis. This form is written as \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.
After starting from the point-slope form \( y - 1 = \frac{1}{4}x - 1 \) and simplifying, we reach \( y = \frac{1}{4}x \). Here, the slope \( m \) remains \( \frac{1}{4} \), but since there is no added constant, the y-intercept \( b \) is \( 0 \).
Examples of how it is useful:
  • Quick Interpretation: This form shows you at a glance how steep the line is and exactly where it cuts through the y-axis.
  • Graphing Clarity: Plotting is straightforward: start at the y-intercept \( b \), then use the slope to find other points.
This format is highly practical when dealing with linear graph equations and predictions.
Standard form of a line
The standard form of a line's equation is handy for various mathematical and practical applications. It is written as \( Ax + By + C = 0 \), where \( A \), \( B \), and \( C \) are integers with \( A \) usually positive.
In our example, we convert from the slope-intercept form \( y = \frac{1}{4}x \) by eliminating fractions and rearranging terms: multiply all parts by 4 to get \( 4y = x \). Rearrange to get \( -x + 4y = 0 \), often written as \( x - 4y = 0 \).
Advantages of this form include:
  • Universal Comparisons: It makes it easy to compare differences or similarities between linear equations.
  • Clear Coefficients: Having integer coefficients simplifies certain calculations and can help in programming or formulaic contexts.
Understanding standard form not only aids computations but also assists in comprehending more complex mathematical concepts.

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Most popular questions from this chapter

In Problems 45 -48, find the coordinates of the point of intersection. Then write an equation for the line through that point perpendicular to the line given first. \(4 x-5 y=8\) \(-3 x+y=5\) \(2 x+y=-10\)

Calculate (be sure that your calculator is in radian or degree mode as needed). (a) \(\frac{56.4 \tan 34.2^{\circ}}{\sin 34.1^{\circ}}\) (b) \(\frac{5.34 \tan 21.3^{\circ}}{\sin 3.1^{\circ}+\cot 23.5^{\circ}}\) (c) \(\tan 0.452\) (d) \(\sin (-0.361)\)

Show that the set of points that are twice as far from \((3,4)\) as from \((1,1)\) form a circle. Find its center and radius.

Sketch the graphs of the following on \([-\pi, 2 \pi]\). (a) \(y=\csc t\) (b) \(y=2 \cos t\) (c) \(y=\cos 3 t\) (d) \(y=\cos \left(t+\frac{\pi}{3}\right)\)

Does \((3,9)\) lie above or below the line \(y=3 x-1\) ?
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