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Chapter 9: Problem 32

Solve each system. State whether it is inconsistent or has infinitely many solutions. If the system has infinitely many solutions, write the solution set with y arbitrary. $$\begin{aligned} &3 x+2 y=5\\\ &6 x+4 y=8 \end{aligned}$$

Short Answer

Expert verified
The system is inconsistent.

Step by step solution

01

Write down the system of equations

The system of equations given is: 1) \(3x + 2y = 5\) 2) \(6x + 4y = 8\)
02

Compare the equations

Notice that the second equation is a multiple of the first equation. Dividing the second equation by 2, we get: \(\frac{6x + 4y}{2} = \frac{8}{2}\) Simplifying, we get: \(3x + 2y = 4\)
03

Identify the inconsistency

Now we have two equations: 1) \(3x + 2y = 5\) 2) \(3x + 2y = 4\) Since \(5 eq 4\), it is clear that these two equations contradict each other, meaning there is no solution that satisfies both equations simultaneously.
04

Conclude the nature of the system

Since the two equations contradict one another, the system of equations is inconsistent.

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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 expression in which each term is either a constant or the product of a constant and a single variable. These equations form a straight line when graphed on a Cartesian plane. Linear equations typically look like this:
  • Two variables: ax + by = c
  • One variable: ax = c
In the given system, both equations - 3x + 2y = 5 and 6x + 4y = 8 - are linear equations that involve two variables, x and y. Understanding how these equations behave in a coordinate system is crucial for solving systems of equations.
solving systems
Solving systems of linear equations means finding the values of the variables that satisfy all equations simultaneously. There are various methods to solve these systems, including:
  • Graphing: Plotting each equation on a graph to find the intersection points
  • Substitution: Solving one equation for a variable and substituting this value into the other equation
  • Elimination: Adding or subtracting equations to eliminate one variable
In the provided exercise, the elimination method is implicitly used. By comparing the given equations, we see that the second equation is a multiple of the first. Dividing the second equation by 2 simplifies it to the first equation, suggesting that the two equations should coincide. However, an inconsistency arises because their constants do not match after simplification.
algebraic inconsistency
Algebraic inconsistency occurs when a system of equations cannot be satisfied by any set of variable values. In simpler terms, no solution exists that makes both equations true at the same time. The given exercise demonstrates this concept well:
Start with the system: 3x + 2y = 5 6x + 4y = 8
Dividing the second equation by 2 gives:
3x + 2y = 4 This reveals a contradiction: 3x + 2y = 5 鈮� 3x + 2y = 4 Because these simplified equations cannot both be true, we conclude that the system is inconsistent. This means the lines represented by these equations are parallel and never intersect.

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Most popular questions from this chapter

Solve each problem. Purchasing costs The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix \(A\) Eggs lour Sugar Shortening Milk \(\left.\begin{array}{l|ccccc}\text { Rolls (doz) } & 1 & 4 & \frac{1}{4} & \frac{1}{4} & 1 \\ \text { Bread (loaf) } & 0 & 3 & 0 & \frac{1}{4} & 0 \\ \text { Cake } & 4 & 3 & 2 & 1 & 1 \\ \text { Pie (crust) } & 0 & 1 & 0 & \frac{1}{3} & 0\end{array}\right]=A\) The cost (in cents) for each ingredient when purchased in large lots or small lots is given by matrix \(B\) Large Lot Small Lot \(\left.\begin{array}{l|rr}\text { Eggs } & 5 & 5 \\ \text { Flour } & 8 & 10 \\\ \text { Sugar } & 10 & 12 \\ \text { Shortening } & 12 & 15 \\ \text { Milk } & 5 & 6\end{array}\right]=B\) (a) Use matrix multiplication to find a matrix giving the comparative cost per bakery item for the two purchase options. (b) Suppose a day's orders consist of 20 dozen sweet rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a \(1 \times 4\) matrix, and, using matrix multiplication, write as a matrix the amount of each ingredient needed to fill the day's orders. (c) Use matrix multiplication to find a matrix giving the costs under the two purchase options to fill the day's orders.

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} &4 x-3 y+z+1=0\\\ &\begin{array}{l} 5 x+7 y+2 z+2=0 \\ 3 x-5 y-z-1=0 \end{array} \end{aligned}$$

Concept Check Find \(A B\) and \(B A\) for $$ A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$ What do you notice? Matrix \(B\) acts as the multiplicative ___________ element for \(2 \times 2\) square matrices.

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} x+y+z &=4 \\ 2 x-y+3 z &=4 \\ 4 x+2 y-z &=-15 \end{aligned}$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{array}{r} 2 x-y+4 z=-2 \\ 3 x+2 y-z=-3 \\ x+4 y+2 z=17 \end{array}$$
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