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Chapter 1: Problem 3

In each of the following systems, interpret each equation as a line in the plane. For each system, graph the lines and determine geometrically the number of solutions. (a) $$\begin{array}{l} x_{1}+x_{2}=4 \\ x_{1}-x_{2}=2 \end{array}$$ (b) $$\begin{array}{r} x_{1}+2 x_{2}=4 \\ -2 x_{1}-4 x_{2}=4 \end{array}$$ (c) $$\begin{array}{r} 2 x_{1}-x_{2}=3 \\ -4 x_{1}+2 x_{2}=-6 \end{array}$$ (d) $$\begin{aligned} x_{1}+x_{2} &=1 \\ x_{1}-x_{2} &=1 \\ -x_{1}+3 x_{2} &=3 \end{aligned}$$

Short Answer

Expert verified
To summarize, the given systems of linear equations have the following number of solutions: (a) One unique solution (b) One unique solution (c) No solution (d) No solution

Step by step solution

01

Rewrite in slope-intercept form

Rewrite both equations as follows: \(x_{2}=-x_{1}+4\) (1) \(x_{2}=x_{1}-2\) (2)
02

Identify the slope and y-intercept

From (1), we have the slope m=-1 and y-intercept b=4. From (2), we have the slope m=1 and y-intercept b=-2.
03

Graph the lines and analyze their relationship

Plot the lines with their slope and y-intercept. The lines will intersect at a single point, meaning the system has one unique solution.
04

Determine the number of solutions

Since the lines intersect at one point, the system has one unique solution. (b)
05

Rewrite in slope-intercept form

Rewrite both equations as follows: \(x_{2}=-\frac{1}{2}x_{1}+2\) (1) \(x_{2}=\frac{1}{4}x_{1}-1\) (2)
06

Identify the slope and y-intercept

From (1), we have the slope m=-1/2 and y-intercept b=2. From (2), we have the slope m=1/4 and y-intercept b=-1.
07

Graph the lines and analyze their relationship

Plot the lines with their slope and y-intercept. The lines will intersect at a single point, meaning the system has one unique solution.
08

Determine the number of solutions

Since the lines intersect at one point, the system has one unique solution. (c)
09

Rewrite in slope-intercept form

Rewrite both equations as follows: \(x_{2}=2x_{1}-3\) (1) \(x_{2}=2x_{1}+\frac{3}{2}\) (2)
10

Identify the slope and y-intercept

From (1), we have the slope m=2 and y-intercept b=-3. From (2), we have the slope m=2 and y-intercept b=3/2.
11

Graph the lines and analyze their relationship

Plot the lines with their slope and y-intercept. The lines will be parallel, meaning the system has no solution.
12

Determine the number of solutions

Since the lines are parallel, the system has no solution. (d)
13

Rewrite in slope-intercept form

Rewrite all three equations as follows: \(x_{2}=-x_{1}+1\) (1) \(x_{2}=x_{1}-1\) (2) \(x_{2}=\frac{1}{3}x_{1}+1\) (3)
14

Identify the slope and y-intercept

From (1), we have the slope m=-1 and y-intercept b=1. From (2), we have the slope m=1 and y-intercept b=-1. From (3), we have the slope m=1/3 and y-intercept b=1.
15

Graph the lines and analyze their relationship

Plot the lines with their slope and y-intercept. The lines intersect in such a way that there isn't a common intersection point among all three lines, meaning the system has no solution.
16

Determine the number of solutions

Since there isn't a common intersection point among all three lines, the system has no solution.

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Key Concepts

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

Graphical Interpretation
In the context of solving systems of linear equations, graphical interpretation involves understanding each linear equation as a line on a two-dimensional plane. Each line represents all the possible solutions of the equation. When multiple equations are involved, we plot all the lines and look for their intersection points.

An intersection point signifies a possible solution shared by the equations.
  • If all lines intersect at a single point, the system has one solution, called a unique solution.
  • If the lines coincide completely, the system has infinitely many solutions.
  • If lines are parallel and never meet, the system has no solution.
Graphical interpretation provides a visual understanding of how equations relate to one another, helping us identify the number of solutions a system may have.
Slope-Intercept Form
The slope-intercept form is a way to express a linear equation in the format: \[ y = mx + b \] where \( m \) is the slope of the line and \( b \) is the \( y \)-intercept. The \( y \)-intercept is the point where the line crosses the \( y \)-axis.

The slope \( m \) indicates the steepness and direction of the line. A positive slope means the line ascends from left to right, while a negative slope descends.
  • When converting a standard form equation to slope-intercept form, it becomes easier to plot the line.
  • Comparing equations in slope-intercept form helps quickly identify if lines are parallel or intersecting.
Understanding and using slope-intercept form is critical for graphical interpretation of linear equations.
Unique Solution
A unique solution in a system of linear equations is when there is exactly one point that satisfies all equations in the system. This occurs when the lines represented by the equations intersect at a single point.

To have a unique solution, two distinct lines must not be parallel. Instead, they should intersect, meaning they share exactly one coordinate pair that makes each equation true.
  • This solution is important in many practical applications, like finding the intersection of costs vs. revenue in economics.
  • Graphically, a unique solution appears as the intersection point of two non-parallel lines.
Ensuring a unique solution is vital for accurate results in any analysis involving linear systems.
Parallel Lines
Parallel lines are lines in a plane that never intersect, meaning they never share a common point. In terms of linear equations, two lines are parallel if they have the same slope but different \( y \)-intercepts.

Analyzing parallel lines helps us understand systems of equations that have no solution. This is because parallel lines can't have a common intersection point.
  • If equations in a system have identical slopes, they are parallel and won't meet.
  • Graphically, these lines will appear to have uniform distance across the graph, just sliding along the plane.
Recognizing parallel lines ensures that one understands when a system is unsolvable, due to the lines not meeting at any point.

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Most popular questions from this chapter

Let \(\left(c_{1}, c_{2}\right)\) be a solution of the \(2 \times 2\) system \\[ \begin{array}{l} a_{11} x_{1}+a_{12} x_{2}=0 \\ a_{21} x_{1}+a_{22} x_{2}=0 \end{array} \\] Show that for any real number \(\alpha\) the ordered pair \(\left(\alpha c_{1}, \alpha c_{2}\right)\) is also a solution

Find nonzero \(2 \times 2\) matrices \(A\) and \(B\) such that \(A B=O\)

Compute the LU factorization of each of the following matrices. (a) \(\left(\begin{array}{ll}3 & 1 \\ 9 & 5\end{array}\right)\) (b) \(\left(\begin{array}{rr}2 & 4 \\ -2 & 1\end{array}\right)\) (c) \(\left(\begin{array}{rrr}1 & 1 & 1 \\ 3 & 5 & 6 \\ -2 & 2 & 7\end{array}\right)\) (d) \(\left(\begin{array}{rrr}-2 & 1 & 2 \\ 4 & 1 & -2 \\ -6 & -3 & 4\end{array}\right)\)

Let \(A, B, L, M, S,\) and \(T\) be \(n \times n\) matrices with \(A, B,\) and \(M\) nonsingular and \(L, S,\) and \(T\) singular Determine whether it is possible to find matrices \(X\) and \(Y\) such that $$\left(\begin{array}{llllll} O & I & O & O & O & O \\ O & O & I & O & O & O \\ O & O & O & I & O & O \\ O & O & O & O & I & O \\ O & O & O & O & O & X \\ Y & O & O & O & O & O \end{array}\right)\left(\begin{array}{l} M \\ A \\ T \\ L \\ A \\ B \end{array}\right)=\left(\begin{array}{l} A \\ T \\ L \\ A \\ S \\ T \end{array}\right)$$ If so, show how; if not, explain why.

An \(n \times n\) matrix \(A\) is said to be an involution if \(A^{2}=I .\) Show that if \(G\) is any matrix of the form $$G=\left(\begin{array}{rr} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{array}\right)$$ then \(G\) is an involution.
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