/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 61 $$\left|\begin{array}{ll} 2 &a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 61

$$\left|\begin{array}{ll} 2 & 5 \\ 1 & x \end{array}\right|=3$$

Short Answer

Expert verified
The value of \( x \) is 4.

Step by step solution

01

Set up the determinant equation

The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated as \( ad - bc \). For the matrix given in the problem, it is \( 2x - 5 \cdot 1 \). Set this equal to the given determinant value of 3: \( 2x - 5 = 3 \).
02

Solve for x

Now solve the equation \( 2x - 5 = 3 \). Start by adding 5 to both sides of the equation to get \( 2x = 8 \). Then divide both sides by 2 to find \( x = 4 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

2x2 matrix
A 2x2 matrix is a simple type of matrix used often in algebra. It consists of four elements arranged in two rows and two columns. Imagine it as a little box or table with numbers in it. Here鈥檚 a simple representation:
  • The top left and right elements are in the first row.
  • The bottom left and right elements are in the second row.
If you have a matrix like \( \begin{pmatrix} 2 & 5 \ 1 & x \end{pmatrix} \), then:
  • "2" is the top left,
  • "5" is the top right,
  • "1" is the bottom left,
  • "x" is the bottom right element that we often are solving for.
Using this 2x2 setup, we can do various calculations, like finding a determinant.
Basic Algebra
Basic Algebra involves solving equations to find the value of unknown variables. It's kind of like being a detective where you look for clues to figure out a mystery number. In the example problem, you are given an equation derived from the matrix's determinant. It is written as \(2x - 5 = 3\). Here's how you solve it:
  • Start by isolating the variable: You want to have "x" alone on one side of the equation. You do this by manipulating the equation in basic steps.
  • Add 5 to both sides: This helps cancel the -5 on the left, leading to \(2x = 8\).
  • Divide by 2: Finally, to solve for "x," divide both sides by 2. You will find that \(x = 4\).
Each step gets you closer to the solution, showing the power of basic algebraic manipulation.
Matrix Determinant Calculation
The determinant of a matrix gives valuable information and underlies many operations in linear algebra. For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), this is found using a straightforward formula: \(ad - bc\).Let's consider our example:
  • The matrix is \(\begin{pmatrix} 2 & 5 \ 1 & x \end{pmatrix}\).
  • Plug into the formula: Multiply the diagonal top-left to bottom-right: \(2 \times x\).
  • Subtract the product of the other diagonal: \(5 \times 1\).
This simplifies to the determinant calculation \(2x - 5\). In this specific problem, the determinant is set to equal 3. By solving the simple algebraic equation, \(2x - 5 = 3\), it helps to find the unknown value of "x". This straightforward operation helps to answer various mathematical problems involving matrices.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits. The power (in W) dissipated in an electric resistance (in \(\Omega\) ) equals the resistance times the square of the current (in \(\mathrm{A}\) ). If 1.0 A flows through resistance \(R_{1}\) and 3.0 A flows through resistance \(R_{2},\) the total power dissipated is \(14.0 \mathrm{W}\). If \(3.0 \mathrm{A}\) flows through \(R_{1}\) and \(1.0 \mathrm{A}\) flows through \(R_{2},\) the total power dissipated is \(6.0 \mathrm{W}\). Find \(R_{1}\) and \(R_{2}\)

Set up appropriate systems of two linear equations in two unknowns and then solve the systems by determinants. All numbers are accurate to at least two significant digits. A machinery sales representative receives a fixed salary plus a sales commission each month. If \(\$ 6200\) is earned on sales of \(\$ 70,000\) in one month and \(\$ 4700\) is earned on sales of \(\$ 45,000\) in the following month, what are the fixed salary and the commission percent?

Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits. A person invested a total of \(\$ 20,900\) into two bonds, one with an annual interest rate of \(6.00 \%\) and the other with an annual interest rate of \(5.00 \%\) per year. If the total annual interest from the bonds is \(\$ 1170,\) how much is invested in each bond?

Determine the value of \(k\) that makes the system dependent. $$\begin{aligned} &3 x-k y=6\\\ &x+2 y=2 \end{aligned}$$

Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits. A manufacturer produces three models of DVD players in a year. Four times as many of model A are produced as model C. and 7000 more of model \(B\) than model \(C\). If the total production for the year is 97,000 units, how many of each are produced?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.