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Chapter 9: Problem 54

Parameterize the equation of the line given in standard form. $$ x-5 y+7=0 $$

Short Answer

Expert verified
The parametric equations are \(x = t\) and \(y = \frac{1}{5}t + \frac{7}{5}\).

Step by step solution

01

Understand the Standard Form

The standard form of a linear equation in two variables is given as \(Ax + By + C = 0\). In the given equation, \(A=1\), \(B=-5\), and \(C=7\).
02

Solve for y

To parameterize, we first need to express \(y\) in terms of \(x\). Solve the equation \(x - 5y + 7 = 0\) for \(y\). Move \(x\) and \(7\) to the other side: \(-5y = -x - 7\). Now divide every term by \(-5\), resulting in \(y = \frac{1}{5}x + \frac{7}{5}\).
03

Parameterize the Equation using a Parameter t

We introduce a parameter \(t\) to express both \(x\) and \(y\) in terms of \(t\). Set \(x = t\). Using the expression for \(y\) from Step 2, substitute \(x\) with \(t\): \(y = \frac{1}{5}t + \frac{7}{5}\).
04

Write the Parametric Equations

The parameterized equations are: \(x = t\) and \(y = \frac{1}{5}t + \frac{7}{5}\), where \(t\) is any real number.

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Key Concepts

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

Linear Equations and Their Forms
Linear equations form the foundation of algebra, representing relationships where each variable occurs to the first power, and their graph is a straight line. The most common forms of a linear equation are:

  • Standard Form: This is written as \(Ax + By + C = 0\). Here, \(A\), \(B\), and \(C\) are constants, and \(x\), \(y\) are the variables.
  • Slope-Intercept Form: This is expressed as \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
In any form, a linear equation describes a constant rate of change, which is depicted as a straight line on a graph. Understanding each form allows us to solve, convert, and analyze the equation for different applications.
Parametric Equations Simplified
Parametric equations provide a flexible way to express the equations of lines, focusing on the role of a parameter. Unlike the standard or slope-intercept forms, parametric equations introduce a third variable, usually \(t\), to define the x and y coordinates. This is particularly useful when:

  • Representing motion: Where each variable depends on time \(t\).
  • Simplifying complex curves or transforming equations for easier computation.
To parameterize a line, two equations are established such that \(x = f(t)\) and \(y = g(t)\), effectively linking the line's position to the parameter \(t\). It translates the problem of finding points on the line into finding the appropriate \(t\) values, enhancing the comprehension of geometric shapes.
Coordinate Geometry and Lines
Coordinate geometry, or analytic geometry, utilizes an ordered pair of numbers (coordinates) to pinpoint the location of points on a plane. It elegantly merges algebra and geometry, allowing for the exploration of geometric shapes through algebraic equations. Through coordinate geometry, we analyze:

  • Points: Defined by coordinates \((x, y)\).
  • Lines: Expressed algebraically, such as in parametric form, which makes understanding shifts, stretches, and rotations intuitive.
  • Distance and Midpoints: Calculated using coordinate principles to determine lengths and origins swiftly.
This methodology revolutionizes our understanding of shapes and their properties, providing a visual representation that is considerably more insightful than numerical calculations alone. It gives a graphical consistency to real-world tasks, from design to physics and beyond.

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Most popular questions from this chapter

Find the inverse matrix to each given matrix if the inverse matrix exists. $$ A=\left[\begin{array}{rrr} -1 & 0 & -1 \\ 0 & -2 & 0 \\ -1 & 1 & 2 \end{array}\right] $$

Zach wants to buy fish and plants for his aquarium. Each fish costs \(\$ 2.30 ;\) each plant costs \(\$ 1.70 .\) He buys a total of 11 items and spends a total of \(\$ 21.70 .\) Set up a system of linear equations that will allow you to determine how many fish and how many plants Zach bought, and solve the system.

Suppose that $$ A=\left[\begin{array}{ll} a & 8 \\ 2 & 4 \end{array}\right], \quad X=\left[\begin{array}{l} x \\ y \end{array}\right], \quad \text { and } \quad B=\left[\begin{array}{l} b_{1} \\ b_{2} \end{array}\right] $$ (a) Show that when \(a \neq 4, A X=B\) has exactly one solution. (b) Suppose \(a=4 .\) Find conditions on \(b_{1}\) and \(b_{2}\) such that \(A X=\) \(B\) has (i) infinitely many solutions and (ii) no solutions. (c) Explain your results in (a) and (b) graphically.

Find the augmented matrix and use it to solve the system of linear equations. $$ \begin{aligned} -x-2 y+3 z &=-9 \\ 2 x+y-z &=5 \\ 4 x-3 y+5 z &=-9 \end{aligned} $$

Write each system in matrix form. $$ \begin{array}{r} 2 x_{1}+3 x_{2}-x_{3}=0 \\ 2 x_{2}+x_{3}=1 \\ x_{1} \quad-2 x_{3}=2 \end{array} $$
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