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Chapter 2: Problem 3

a. \(2 x-4 y+z=1\) \(4 x+2 y-z=1\)

Short Answer

Expert verified
From the solution process, we have two equations representing our solution: \(z = x + 3y\) and \(3x - y = 1\). There isn't a unique solution for each variable because we need at least as many equations as variables.

Step by step solution

01

Step 1. Equate the equations

To simplify the system, let's equate the two equations. We will subtract second equation from the first one: \( (2x - 4y + z) - (4x + 2y - z) = 1 - 1 \)
02

Step 2. Simplify the result

Perform the subtraction operation: \(-2x - 6y + 2z = 0\). this can be simplified by dividing everything by -2 which results into: \(x + 3y - z = 0\)
03

Step 3. Get value of z

From the last step, we can write z in terms of x and y: \( z = x + 3y \)
04

Step 4. Substitute z in first equation

Substitute z into the first equation: \(2x - 4y + x + 3y = 1\). This simplifies to: \(3x - y = 1\)

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Key Concepts

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

Systems of Linear Equations
A System of Linear Equations is a collection of two or more linear equations involving the same set of variables. In a system, each equation conveys a relationship between the variables, and our objective is to find values of these variables that satisfy all equations simultaneously.
For example, consider the following system of equations:
  • Equation 1: \(2x - 4y + z = 1\)
  • Equation 2: \(4x + 2y - z = 1\)
In this system, we aim to find a set of values for \(x\), \(y\), and \(z\) such that both equations hold true together.
The beauty of these systems is that they can have different types of solutions: a single unique solution, infinitely many solutions, or no solution at all. Our task is to unravel the system step by step to determine which type of solution applies.
Equation Simplification
Equation Simplification is an essential skill in solving systems of linear equations. It involves manipulating equations to make them easier to work with by reducing complexity.
In our example, before we can effectively solve the system, it's helpful to simplify it. This can be done by using basic algebraic operations like addition, subtraction, multiplication, or division.
Take the subtraction operation in our case:
  • We subtract the entire second equation from the first equation: \((2x - 4y + z) - (4x + 2y - z) = 1 - 1\).
  • Performing the subtraction simplifies to \(-2x - 6y + 2z = 0\).
    • To make it simpler, divide the equation by \(-2\), resulting in \(x + 3y - z = 0\).
By simplifying equations, we reduce complexity and make it easier to identify relationships or patterns between variables.
Substitution Method
The Substitution Method is a common technique for solving systems of linear equations. It involves isolating one variable in one equation and substituting it into another equation to solve for the remaining variables.
In our example, after simplifying the system, we derived the equation \(x + 3y - z = 0\).
  • First, isolate \(z\) from this equation: \(z = x + 3y\).
  • Next, replace \(z\) in the first original equation with \(x + 3y\):
    Thus, it becomes \(2x - 4y + (x + 3y) = 1\).
  • Simplify to get \(3x - y = 1\).
This step helps us reduce the system further and directly find values for some of the variables, making it a powerful strategy in solving linear equations.

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Most popular questions from this chapter

(Method of undetermined coefficients) Find values of the coefficients \(a, b\), and \(c\) so that $$ y=a e^{x}+b x e^{x}+c x^{2} e^{x} $$ is a solution to the differential equation $$ y^{\prime}+y=e^{x}+x e^{x}+x^{2} e^{x} . $$

a. Show that the solution set of the equation \(a x+b y+c z=d\) is a plane in \(\mathrm{R}^{3}\) as long as one or more of the coefficients \(a, b, c\) is nonzero. b. Under what conditions on the coefficients does this plane pass through the origin? c. What are the possible solution sets if \(a=b=c=0\) ?

a. \(4 x-6 y+\frac{2}{y} z=2\) \(6 x-9 y+z=3\)

How many columns of the matrix $$ \left[\begin{array}{llll} 1 & 0 & 1 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] $$ contain leading 1 s? How many rows of this matrix contain leading 1 s? If your answers to these two questions are different, be sure to check very carefully the definition of leading 1s.

Put the following matrices in reduced row-echelon fotm. a. \(\quad\left[\begin{array}{rr}2 & -4 \\ -3 & 6 \\ 1 & 2 \\ -2 & 4\end{array}\right]\) b. \(\left[\begin{array}{llll}0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0\end{array}\right]\) c. \(\left[\begin{array}{rrrr}0 & 1 & 2 & 0 \\ -2 & 0 & -6 & -1 \\ 4 & -1 & 10 & 2 \\ 1 & 0 & 3 & 0\end{array}\right]\) d. \(\left[\begin{array}{rrrrrr}0 & 1 & 4 & 1 & 1 & 1 \\ 0 & -1 & -4 & 1 & -1 & 3 \\ 0 & 1 & 4 & 0 & 1 & -1 \\ 0 & 2 & 8 & 3 & 2 & 4\end{array}\right]\)
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