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Chapter 4: Problem 42

Find an equation of the line containing the two given points. Express your answer in the indicated form. \((2,1)\) and \((4,6) ;\) standard form

Short Answer

Expert verified
The equation of the line containing the two given points, (2,1) and (4,6), in standard form is \(5x - 2y = 8\).

Step by step solution

01

Find the Slope

To find the slope of the line, we will use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two given points. Plugging in the values of the points \((2,1)\) and \((4,6)\), we get: \(m = \frac{6 - 1}{4 - 2} = \frac{5}{2}\) So, the slope of the line is \(\frac{5}{2}\).
02

Use Point-Slope Form

With the slope calculated in Step 1, we can now use the point-slope form to find the equation of the line. The point-slope form is given by: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is one of the given points and \(m\) is the slope. Using the point \((2,1)\) and the slope \(\frac{5}{2}\), we have: \(y - 1 = \frac{5}{2}(x - 2)\)
03

Convert to Standard Form

To convert the equation to standard form, we need to eliminate the fractions and have the terms arranged in the form \(Ax + By = C\). Start by multiplying both sides of the equation by \(2\) to eliminate the fraction: \(2(y - 1) = 5(x - 2)\) Distribute the numbers on both sides: \(2y - 2 = 5x - 10\) Now, move the terms with variables to the left side of the equation and the constant terms to the right side: \(-5x + 2y = -8\) The equation is now in standard form, though it's preferable to have the leading coefficient positive. Multiply the equation by \(-1\) to make coefficient of \(x\) positive: \(5x - 2y = 8\) The equation of the line containing the two given points in standard form is \(\boxed{5x - 2y = 8}\).

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Key Concepts

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

Slope Formula
Finding the slope of a line is crucial in understanding its inclination or steepness between two points. The slope formula is expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This simple mathematical expression allows us to measure the vertical change (rise) over the horizontal change (run) between two coordinates.
  • For the points \((2, 1)\) and \((4, 6)\), substitute these into the formula to find the slope \(m\).
  • The coordinates show us that \(x_1 = 2\), \(y_1 = 1\), \(x_2 = 4\), and \(y_2 = 6\). By substituting, we get \( m = \frac{6 - 1}{4 - 2} = \frac{5}{2} \).
  • This calculation tells us that for every 2 units moved horizontally, the line moves 5 units vertically upward.
Understanding slope is fundamental in coordinate geometry, as the slope value \( \frac{5}{2} \) helps us derive other line equations.
Point-Slope Form
Once we have the slope, we can use it along with any point on the line to express the equation of the line in point-slope form. The point-slope form is given by \( y - y_1 = m(x - x_1) \). It is especially useful when a point and a slope are known.
  • Picking the point \((2, 1)\) and slope \( \frac{5}{2} \), the equation becomes \( y - 1 = \frac{5}{2}(x - 2) \).
  • This form makes it easy to visualize how the line starts at a specific point and extends based on its slope.
  • The resulting equation describes how the value of \(y\) changes with \(x\), moving from the known point \((x_1, y_1)\).
Using point-slope form is a versatile step in shifting to other linear equation forms, such as standard form.
Standard Form
Standard form expresses the equation of a line as \( Ax + By = C \), where \(A\), \(B\), and \(C\) are integers, and \(A\) is typically non-negative. Using it helps make comparisons easier with other lines or fitting into coordinate systems.
  • Start by multiplying both sides of the point-slope equation \( y - 1 = \frac{5}{2}(x - 2) \) by 2 to eliminate the fraction: \( 2(y - 1) = 5(x - 2) \).
  • Distribute to get \( 2y - 2 = 5x - 10 \).
  • Rearrange it into standard form \( -5x + 2y = -8 \) which simplifies to \( 5x - 2y = 8 \) by multiplying through by \(-1\).
This rearrangement adds clarity when engaging in further analytical geometry tasks, helping identify parallel or perpendicular lines.
Coordinate Geometry
Coordinate geometry, often known as analytic geometry, is the study of geometry using a coordinate system. It pairs algebra and geometry through graphs and coordinates, providing a clear graphical representation of equations. This field of mathematics is crucial for understanding the spatial relationships between different forms.
  • The coordinate plane consists of two axes (x-axis and y-axis) that divide the plane into four quadrants.
  • Points, lines, and curves are plotted, transforming abstract equations into visual graphs.
  • The slope of a line elucidates the angle it forms with the axes, indicating inclination and direction.
Through problems like finding line equations, coordinate geometry empowers you to visualize mathematical concepts, making complex problems easier to solve and understand.

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Most popular questions from this chapter

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{8}{2 x+1}$$

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. \(y=-3 x-1 ;(0,5) ;\) slope-intercept form

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. $$y=-\frac{4}{3} x+2 ;(8,1) ; \text { slope-intercept form }$$

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

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. $$24 x-15 y=10 ;(16,-7) ; \text { standard form }$$
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