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5 Edition

Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan

Chapter 5: Chapter 5

Problem 1

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Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. $\begin{aligned} x-3 y &=-1 \\ 4 x+3 y &=11 \end{aligned}$

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The given system of linear equations has one and only one solution, which is $(x,y) = (2,1)$.

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TABLE OF CONTENTS

Step 1: Combine the equations to eliminate a variable

: We will first try to eliminate the variable y from both equations. To do this, add both equations: \[x - 3y = -1\] \[4x + 3y = 11\] Add both equations: \[(x - 3y) + (4x + 3y) = -1 + 11\]

Step 2: Simplify the combined equation

: Now, simplify the combined equation to find the value of `x`: \[1x + 4x = 10\] \[5x = 10\] Now, divide both sides by 5: \[\frac{5x}{5} = \frac{10}{5}\] \[x = 2\]

Step 3: Substitute the value of variable `x` in one of the equations

: Now that we have the value of `x`, we can substitute this value into either of the equations to find the value of `y`. We'll use the first equation: \[x - 3y = -1\] \[2 - 3y = -1\]

Step 4: Solve for `y`

: Now, solve for `y`: \[3y = 2 + 1\] \[3y = 3\] Divide both sides by 3: \[\frac{3y}{3} = \frac{3}{3}\] \[y = 1\]

Step 5: Finalize the results

: We have found unique values for both x and y: \[x = 2\] \[y = 1\] Since we have found unique values for both variables, the given system of linear equations has one and only one solution. The single and unique solution to the system is: \[(x,y) = (2,1)\]

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Chapter 5: Chapter 5

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\begin{aligned} x-3 y &=-1 \\ 4 x+3 y &=11 \end{aligned}\)

Short Answer

Step by step solution

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Step 1: Combine the equations to eliminate a variable

Step 2: Simplify the combined equation

Step 3: Substitute the value of variable `x` in one of the equations

Step 4: Solve for `y`

Step 5: Finalize the results

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More chapters from the book ‘Applied Mathematics: For the Managerial, Life, and Social Sciences’

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

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