Cramer's Rule - 2x2 Linear System

The Organic Chemistry Tutor
19 Feb 201810:32
EducationalLearning
32 Likes 10 Comments

TLDRThis script offers a comprehensive guide on solving systems of linear equations with two variables using Kramer's Rule. It explains the process step by step, starting with identifying coefficients and constants, to calculating the determinant (D), and finally finding the values of x and y through the determinants D_x and D_y. Two examples are worked out in detail, demonstrating how to apply the rule and verify the solutions. The explanation is clear, making it an excellent resource for those looking to understand and practice this mathematical method.

Takeaways
  • 📚 Kramer's Rule is a method for solving a system of linear equations with two variables.
  • 🔢 The system is set up in the form ax + by = c, where a, b, and c are coefficients and constants.
  • 🌟 The determinant (d) is used to calculate the unique solution for x and y, if it's non-zero.
  • 🛠️ To calculate d, use the formula d = (a1 * b2) - (b1 * a2) for a 2x2 matrix.
  • 📐 The determinant for dx (dx) is calculated by replacing the x coefficients with the constants (c1 and c2) and vice versa for dy (dy).
  • 📈 The value of x is found by dividing dx by d, and y is found by dividing dy by d.
  • 📌 Example 1: For the system 2x + 5y = 26 and 5x - 4y = -1, the solution is x = 3, y = 4.
  • 📌 Example 2: For the system 3x - 2y = -4 and 4x - y = 3, the solution is x = 2, y = 5.
  • 🔄 To check the solution, plug the values of x and y back into the original equations to ensure they hold true.
  • 💡 Kramer's Rule is particularly useful when the system of equations is consistent and has a unique solution.
Q & A

  • What is the primary focus of this lesson?

    -The primary focus of this lesson is to demonstrate how to use Kramer's Rule to solve a system of linear equations with two variables.

  • What are the two equations given in the lesson?

    -The two equations given are 2x + 5y = 26 and 5x - 4y = -1.

  • What does the term 'determinant' refer to in the context of this lesson?

    -In this context, 'determinant' refers to a specific value calculated from a square matrix, which is used in Kramer's Rule to solve for the variables in a system of linear equations.

  • What are the values of a1, a2, b1, b2, c1, and c2 in the first system of equations?

    -For the first system, a1 is 2, a2 is 5, b1 is 5, b2 is 4, c1 is 26, and c2 is -1.

  • How is the determinant (d) calculated for a 2x2 matrix?

    -The determinant for a 2x2 matrix is calculated as (a1 * b2) - (b1 * a2).

  • What is the calculated value of the determinant (d) for the first system of equations?

    -The calculated value of the determinant (d) for the first system is -33.

  • What are the components of the matrix dx in the first system?

    -The components of matrix dx for the first system are c1, c2, b1, and b2, which are 26, -1, 5, and -4 respectively.

  • What is the calculated value of dx for the first system of equations?

    -The calculated value of dx for the first system is -99.

  • How are the values of x and y determined using Kramer's Rule?

    -The values of x and y are determined by dividing dx and dy by the determinant d. x = dx/d and y = dy/d.

  • What is the solution for the first system of equations using Kramer's Rule?

    -The solution for the first system is x = 3 and y = 4.

  • What are the two equations given in the practice example?

    -The two equations given in the practice example are 3x - 2y = -4 and 4x - y = 3.

  • What is the calculated value of the determinant (d) for the practice example?

    -The calculated value of the determinant (d) for the practice example is 5.

  • What is the solution for the practice example using Kramer's Rule?

    -The solution for the practice example is x = 2 and y = 5.

Outlines
00:00
📚 Introduction to Solving Linear Equations with Kramer's Rule

This paragraph introduces the concept of using Kramer's Rule to solve a system of linear equations with two variables. The lesson presents a specific problem with the equations 2x + 5y = 26 and 5x - 4y = -1, and explains the setup of the system in terms of coefficients (a1, a2, b1, b2) and constants (c1, c2). It details the process of calculating the determinant (d), which is key to finding the values of x and y. The explanation includes the step-by-step procedure for finding d, dx, and dy, and concludes with the calculation of x = 3 and y = 4 as the solution to the given equations.

05:01
🔢 Application of Kramer's Rule with a New Example

This paragraph further illustrates the application of Kramer's Rule by introducing a new set of linear equations: 3x - 2y = -4 and 4x - y = 3. The explanation follows a similar structure to the previous paragraph, outlining the calculation of the determinant (d), dx, and dy for this new system. It emphasizes the process of replacing coefficients with constants when calculating dx and dy. The paragraph concludes with the solution to the equations, finding x = 2 and y = 5, and suggests verifying the solution by plugging the values back into the original equations.

10:01
👍 Conclusion on Mastering Kramer's Rule

In this concluding paragraph, the script summarizes the lesson on using Kramer's Rule to solve systems of linear equations with two variables. It reinforces the method's effectiveness and encourages the learner to apply this knowledge to solve similar problems. The paragraph serves as a final recap, highlighting the key points and the successful resolution of the practice examples.

Mindmap
Keywords
💡Kramer's Rule
Kramer's Rule is a method used to solve a system of linear equations with two variables. It is based on the concept of determinants, and it allows us to find the values of the variables by calculating the determinant of the coefficient matrix and the determinants of the matrices formed by replacing the variable's coefficients with the constants from the equations. In the video, Kramer's Rule is used to solve two different systems of linear equations, demonstrating its application in finding the values of x and y.
💡Linear Equations
Linear equations are mathematical equations in which the highest power of the variable is one, and the equation is a straight line when graphed. In the context of this video, the linear equations are those that include two variables, x and y, and are to be solved using Kramer's Rule. The video provides two examples of such systems of equations, each with its own pair of linear equations to be solved.
💡Determinant
A determinant is a scalar value that can be computed from the elements of a square matrix and is used in various areas of mathematics, including linear algebra. In the context of the video, the determinant is crucial for applying Kramer's Rule, as it helps to find the unique solution to a system of linear equations. The determinant is calculated by multiplying the elements along the main diagonal and subtracting the product of the elements along the other diagonal in a 2x2 matrix.
💡Coefficients
Coefficients are numerical factors that are multiplied by variables in a linear equation. They play a crucial role in defining the slope or the inclination of the line represented by the equation. In the video, coefficients are the numbers in front of the variables x and y in the given linear equations. These coefficients are used in the calculation of the determinants required for solving the system using Kramer's Rule.
💡Variables
Variables are symbols, often letters, that represent unknown quantities in a mathematical equation. In the context of this video, the variables are x and y, which are the unknowns being solved for in the system of linear equations. The goal is to find the values of x and y that satisfy both equations simultaneously.
💡Matrix
A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In linear algebra, matrices are used to represent systems of linear equations. The video discusses the use of a 2x2 matrix to apply Kramer's Rule, where the coefficients of the variables are replaced with the constants from the equations to form the matrices needed for calculating the determinants.
💡Two-Variable System
A two-variable system refers to a set of linear equations that have two unknowns, typically represented by the variables x and y. These systems can be represented graphically as the intersection of two lines in a coordinate plane. The video focuses on solving such systems using Kramer's Rule, which provides a unique solution when the determinant of the coefficient matrix is non-zero.
💡Solving Equations
Solving equations refers to the process of finding the values of the unknowns that make the equation true. In the context of this video, solving equations involves applying a specific mathematical method, Kramer's Rule, to find the values of x and y that satisfy the given systems of linear equations.
💡Substitution
Substitution is a mathematical technique used to solve for an unknown by replacing it with an equivalent expression or value. In the video, substitution is used when calculating the determinants for dx and dy by replacing the coefficients of x and y with the constants from the equations.
💡Practice
Practice refers to the act of applying a learned skill or technique through repetition and application in order to improve understanding and proficiency. In the video, practice is emphasized by walking through the process of solving two different systems of linear equations using Kramer's Rule, allowing viewers to follow along and apply the method themselves.
💡Verification
Verification is the process of checking the accuracy or correctness of a solution or result. In the context of the video, verification involves plugging the calculated values of x and y back into the original equations to ensure that they satisfy both equations in the system.
Highlights

Introduction to Kramer's Rule for solving a system of linear equations with two variables.

Given system of equations: 2x + 5y = 26 and 5x - 4y = -1.

Identification of coefficients: a1 is 2, a2 is 5, b1 is 5, b2 is 4, c1 is 26, c2 is -1.

Explanation of determinant (d) calculation for 2x2 system.

Calculation of determinant d as -33 for the given system.

Formula for dx and dy in terms of determinant and coefficients.

Computation of dx as -99 for the given system.

Computation of dy as -132 for the given system.

Solution for x and y using Kramer's Rule: x = 3, y = 4.

Second example provided: 3x - 2y = -4 and 4x - y = 3.

Calculation of determinant d as 5 for the second system.

Computation of dx as 10 for the second system.

Computation of dy as 25 for the second system.

Solution for the second system: x = 2, y = 5.

Verification of solutions by plugging them back into the original equations.

Conclusion summarizing the application of Kramer's Rule for solving two-variable linear systems.

Transcripts

Browse More Related Video

PreCalculus - Matrices & Matrix Applications (32 of 33) Using Cramer's Rule to Find x=? y=?

Cramer's Rule - 2x2 & 3x3 Matrices - Solving Systems of Linear Equations - 2 & 3 Variables

Cramer's Rule - 3x3 Linear System

Substitution Method For Solving Systems of Linear Equations, 2 and 3 Variables, Algebra 2

7.1.1 Solving a System of Equations Using Substitution

Elimination Method For Solving Systems of Linear Equations Using Addition and Multiplication, Algebr

Rate This

5.0 / 5 (0 votes)

Thanks for rating:

Related Tags
Kramer's RuleLinear EquationsMath TutorialTwo VariablesEducational ContentProblem SolvingMathematicsAlgebraTech-Enabled Learning