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Chapter 1: Problem 8

Write \(\left\\{(x, y) \in \mathbb{R}^{2} \mid-3 x+4 y=1\right\\}\) as a line in \(\mathbb{R}^{2}\).

Short Answer

Expert verified
The line in \(\mathbb{R}^{2}\) for the given equation set is \(y = \frac{3}{4}x + \frac{1}{4}\).

Step by step solution

01

Understand and Rewrite the Equation

The first step involves understanding the given expression and then rewrite \(-3x + 4y = 1\) as it reads.
02

Solve for y

Isolate 'y' by rearranging the equation. This is done by adding \(3x\) to each side and then dividing every term by 4. It results in the equation as \(y = \frac{3}{4}x + \frac{1}{4}\).
03

Identify the Line Equation

The equation \(y = \frac{3}{4}x + \frac{1}{4}\) is the desired linear equation with the slope of \(\frac{3}{4}\) and the y-intercept of \(\frac{1}{4}\).

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Key Concepts

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

Linear Equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are fundamental in algebra and they represent lines in two-dimensional spaces, like the Cartesian coordinate system. To visualize them, you plot all the solutions on a graph, which will always form a straight line.

One common form of a linear equation is the slope-intercept form, expressed as y = mx + b, where m is the slope of the line and b is the y-intercept. This form makes it easy to graph the equation and to understand how the line behaves. For example, if you have an equation expressed as -3x + 4y = 1, it can be rearranged into slope-intercept form to easily identify both the slope and y-intercept.
Cartesian Coordinate System
The Cartesian coordinate system is a two-dimensional system used to define locations on a plane through an ordered pair of numbers, commonly written as (x, y). The 'x' represents the horizontal axis, while 'y' represents the vertical axis. These axes intersect at a point called the origin, labeled as (0, 0).

Each point within this system corresponds to one unique position on the plane. Learning how to plot points and lines in the Cartesian coordinate system is crucial for students to visualize mathematical concepts and solve problems that involve spatial relationships. When dealing with linear equations, this coordinate system allows us to create a graphical representation of the equation, making it easier to understand the concept of slope and y-intercept.
Isolating Variables
In algebra, isolating a variable means rewriting an equation so that the variable of interest is alone on one side of the equation. This is a critical step when solving for a variable, and it usually involves performing the same operation on both sides of an equation to maintain the balance.

For instance, if we start with -3x + 4y = 1 and we want to isolate y, we add 3x to each side of the equation, resulting in 4y = 3x + 1. Next, we divide everything by 4 to get y by itself, ending up with y = 3/4x + 1/4. This operation is crucial because it puts the linear equation into slope-intercept form, which is much easier to graph and interpret.
Slope of a Line
The slope of a line is a measure of its steepness and its direction. It is defined as the ratio of the vertical change between two points (rise) to the horizontal change between the same two points (run). In the slope-intercept form y = mx + b, the slope is represented by m.

A positive slope means that the line is inclining upwards as it moves from left to right, while a negative slope indicates it is declining. If a line has a slope of zero, it is horizontal and if the slope is undefined, the line is vertical. For example, the equation derived from -3x + 4y = 1, which is y = 3/4x + 1/4, has a slope m of 3/4. This tells us that for each unit we move to the right on the x-axis, we move up 3/4 units on the y-axis.

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Most popular questions from this chapter

Show that $$ S=\left\\{\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \in \mathbb{M}(2,2) \mid a=b \text { and } b+2 c=0\right\\} $$ is a subspace of \(M(2,2)\).

Prove that \(\frac{1}{2} \mathbf{v}+\frac{1}{2} \mathbf{v}=\mathbf{v}\)

Is it possible to add two sets? For sets of real numbers, we might define addition of sets in terms of addition of the elements in the sets. Let us introduce the following meaning to the symbol \(\oplus\) for adding a set \(A\) of real numbers to another set \(B\) of real numbers: \(A \oplus B=\\{a+b \mid a \in A\) and \(b \in B\\}\). a. List the elements in the set \(\\{1,2,3\\} \oplus\\{5,10\\}\). b. List the elements in the set \(\\{1,2,3\\} \oplus\\{5,6\\}\). c. List the elements in the set \(\\{1,2,3\\} \oplus \varnothing\). d. If a set \(A\) contams \(m\) real numbers and a set \(B\) contains \(n\) real numbers, can you predict the number of elements in \(A \oplus B\) ? If you run into difficulties, can you determine the minimum and naximum numbers of elements possible in \(A \oplus B\) ? e. Does the commutative law hold for this new addition? That is, does \(A \oplus\) \(B=B \oplus A\) ? f. Reformulate other laws of real-number addition in terms of this new addition of sets. Which of your formulas are true? Can you prove them or provide counterexamples? g. What about laws that combine set addition with union and intersection? For example, does \((A \cup B) \oplus C=(A \oplus C) \cup(B \oplus C)\) ? h. Is there any hope of extending the other operations of arithmetic to sets of real numbers? What about algebra? Limits? Power series?

Write \(\left\\{(x, y, z) \in \mathbb{R}^{3} \mid x+y+z=3\right.\) and \(\left.z=2\right\\}\) as a line in \(\mathbb{R}^{3}\).

For each of the following, either explain why the rule gives a well-defined function or find a number that has two values assigned to it. a. \(f: \mathbb{R} \rightarrow \mathbb{R}\) where \(f(x)\) is the third digit to the right of the decimal point in the decimal expansion of \(x\). b. \(g: \mathbb{R} \rightarrow \mathbb{R}\) defined by $$ g(x)= \begin{cases}0 & \text { if } x \text { is irrational } \\ p+q & \text { if } x=\frac{p}{q}, \text { where } p \text { and } q \text { are integers. }\end{cases} $$ c. \(h: \mathbb{R} \rightarrow \mathbb{R}\) defined by $$ h(x)= \begin{cases}0 & \text { if } x \text { is irrational } \\ m-n & \text { if } x=\frac{2^{m} p}{2^{2} q}, \text { where } p \text { and } q \text { are odd integers and } \\ & m \text { and } n \text { are nonnegative integers. }\end{cases} $$ d. \(s:[-1,1] \rightarrow \mathbb{R}\) where \(y=s(x)\) satisfies the equation \(x^{2}+y^{2}=1\).
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