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Chapter 2: Problem 57

Solve the systems of linear equations over the indicated \(\mathbb{Z}_{p}.\) \(3 x+2 y=1\) over \(\mathbb{Z}_{7}\) \(x+4 y=1\)

Short Answer

Expert verified
The solution is \(x \equiv 3\) and \(y \equiv 3\) over \(\mathbb{Z}_{7}\).

Step by step solution

01

Convert Equations to Modulo 7 Format

First, let's ensure each coefficient and constant in the system of equations is reduced modulo 7:The given system is: \[\begin{align*}3x + 2y &\equiv 1 \pmod{7},\x + 4y &\equiv 1 \pmod{7}.\end{align*}\]Since the coefficients and constants are already in the range of 0 to 6, there is no need for further reduction.
02

Solve for One Variable

Start by solving for one variable in the second equation, which is simpler. We can easily solve for \(x\) from the second equation:\[x \equiv 1 - 4y \pmod{7}.\]
03

Substitute Expression into First Equation

Substitute the expression for \(x\) from the previous step into the first equation:\[3(1 - 4y) + 2y \equiv 1 \pmod{7}.\]This simplifies to:\[3 - 12y + 2y \equiv 1 \pmod{7}.\]Combine like terms:\[3 - 10y \equiv 1 \pmod{7}.\]
04

Simplify and Solve for y

Reduce the terms modulo 7:\[-10y \equiv 1 - 3 \pmod{7},\]which simplifies to:\[-10y \equiv -2 \pmod{7}.\]Since \(-10 \equiv 4 \pmod{7}\), rewrite as:\[4y \equiv -2 \pmod{7}.\]Convert \(-2\) to its positive equivalent modulo 7, which is 5:\[4y \equiv 5 \pmod{7}.\]Multiply by the modular inverse of 4 modulo 7. The inverse is 2 because \(4 \times 2 \equiv 1 \pmod{7}\):\[8y \equiv 10 \pmod{7}.\]Which simplifies to:\[y \equiv 3 \pmod{7}.\]
05

Substitute Back to Solve for x

Using \(y \equiv 3 \pmod{7}\), substitute back into the expression for \(x\):\[x \equiv 1 - 4(3) \pmod{7}.\]Calculate:\[x \equiv 1 - 12 \equiv 1 + 2 \equiv 3 \pmod{7}.\]

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Key Concepts

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

Modulo Arithmetic
In mathematics, modulo arithmetic revolves around the operation of finding the remainder when one number is divided by another. It's a fundamental concept used to simplify complex calculations by working within the boundaries of a specified modulus. In our exercise, we use modulo 7, denoted as \(\mathbb{Z}_{7}.\) This means any number will be reduced to its smallest non-negative number within the range 0 to 6 after performing arithmetic operations.
### Key Points
  • Modulo arithmetic often appears in scenarios where cyclical patterns or periodic repetition are involved, such as clock arithmetic.
  • Operations include addition, subtraction, and multiplication, but results are always taken mod the given number (in this case, 7).
  • Working "modulo" ensures we remain within a finite set of numbers, which simplifies computations and provides a structured way to handle periodic problems.
#### ExampleConsider solving \(3x + 2y \equiv 1 \pmod{7}.\) This implies finding values of \(x \)and \(y \)within the cycle of 0 through 6 that satisfy the equation.
Linear Algebra
Linear Algebra is the branch of mathematics concerning linear equations, matrices, and vector spaces. It provides powerful tools for analyzing systems of linear equations, like the ones in our exercise.
### Systems of Linear Equations
  • A system of linear equations consists of two or more linear equations with common variables. Solving it means finding all possible values of the variables that satisfy each equation simultaneously.
  • Our problem involves two linear equations that require us to find suitable integer solutions for \( x \) and \( y \) in a way where both equations hold true.
### Benefits of Linear Algebra
  • It provides systematic methods for finding these solutions, including substitution, elimination, and the use of matrices.
  • Understanding the principles of linear algebra can help handle complex problems in fields like computer science, physics, and engineering.
In our example, we use substitution to solve for one variable and then plug it into another equation, adhering to the principles of linear algebra.
Modular Inverse
The concept of a modular inverse is crucial in modular arithmetic. It involves finding a number which, when multiplied by a given number, results in a product of 1 mod the chosen modulus.
### How It Works
  • For a number \( a \) and modulus \( m \), the modular inverse is a number \( b \) such that \( a \times b \equiv 1 \pmod{m}. \)
  • In our problem, to effectively solve for \(y \) in the equation \(4y \equiv 5 \pmod{7}, \) we needed the modular inverse of 4 mod 7.
### Finding the Inverse
  • To find the inverse, you often need to try values until you find one that satisfies the condition, or use the extended Euclidean algorithm.
  • In our case, multiplying 4 by 2 gives us 8, which is equivalent to 1 mod 7, confirming that the modular inverse of 4 is 2.
Using this inverse allowed us to effectively isolate and solve for our variable, providing the solution to the system of equations.

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Most popular questions from this chapter

The coefficient matrix is not strictly diagonally dominant, nor can the equations be rearranged to make it so. However, both the Jacobi and the Gauss- Seidel method converge anyway. Demonstrate that this is true of the Gauss- Seidel method, starting with the zero vector as the initial approximation and obtaining a solution that is accurate to within 0.01. $$\begin{aligned} 5 x_{1}-2 x_{2}+3 x_{3} &=-8 \\ x_{1}+4 x_{2}-4 x_{3} &=102 \\ -2 x_{1}-2 x_{2}+4 x_{3} &=-90 \end{aligned}$$

Let \(\mathbf{p}=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right], \mathbf{u}=\left[\begin{array}{r}1 \\ 1 \\ -1\end{array}\right],\) and \(\mathbf{v}=\left[\begin{array}{l}2 \\ 1 \\ 0\end{array}\right] .\) Describe all points \(Q=(a, b, c)\) such that the line through \(Q\) with direction vector \(\mathbf{v}\) intersects the line with equation \(\mathbf{x}=\mathbf{p}+\mathbf{s u}.\)

For what value(s) of \(k,\) if any, will the systems have (a) no solution, (b) a unique solution, and (c) infinitely many solutions? \(\begin{aligned} x+y+z &=2 \\ x+4 y-z &=k \\ 2 x-y+4 z &=k^{2} \end{aligned}\)

Set up and solve an appropriate system of linear equations to answer the questions. The sum of Annie's, Bert's, and Chris's ages is 60 . Annie is older than Bert by the same number of years that Bert is older than Chris. When Bert is as old as Annie is now, Annie will be three times as old as Chris is now. What are their ages?

The systems of equations are nonlinear. Find substitutions (changes of variables) that convert each system into a linear system and use this linear system to help solve the given system. $$\begin{array}{l} -2^{a}+2\left(3^{b}\right)=1 \\ 3\left(2^{a}\right)-4\left(3^{b}\right)=1 \end{array}$$
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