How do we solve a system of linear equations using any method

Brian McLogan
13 Aug 201304:33

TLDRThis tutorial demonstrates solving a system of linear equations using the method of elimination by finding the least common multiple (LCM) of the coefficients. The instructor multiplies the equations to align coefficients of variables, leading to the elimination of one variable. After finding 'y' by solving the simplified equations, 'x' is determined by substituting 'y' back into the original equation. The process results in the coordinates of the intersection point, showcasing a clear and methodical approach to solving such systems.

Takeaways

  • 🔢 To solve a system of linear equations, it's crucial to find a common multiple for the coefficients that allows for elimination.
  • 📚 The least common multiple (LCM) is key in determining what to multiply each equation by to align coefficients.
  • 🤔 For the coefficients 7 and 3, the LCM is 21, requiring the top equation to be multiplied by 3 and the bottom by 7.
  • 🧮 Similarly, for coefficients 5 and -4, the LCM is 20, indicating the top equation should be multiplied by 4 and the bottom by 5.
  • 📐 The choice to eliminate either x or y is at the solver's discretion and won't affect the final result.
  • ➗ After aligning coefficients, apply the distributive property to ensure all terms in the equations are multiplied by the chosen multipliers.
  • 🔄 The resulting equivalent equations after multiplication should have the same coefficients for the variable to be eliminated.
  • ➖ Subtracting the equations with aligned, same-signed coefficients eliminates one variable, making it easier to solve for the other.
  • 🆗 Once the value of one variable is found, substitute it back into one of the original or equivalent equations to solve for the second variable.
  • 📈 The solution to the system of equations is the point of intersection, which in this case is (-1, -1).

Q & A

  • What is the main topic discussed in the transcript?

    -The main topic discussed in the transcript is solving a system of linear equations using the method of multiplying by least common multiples (LCM) to eliminate variables.

  • Why is finding the least common multiple important in solving the system of equations presented?

    -Finding the least common multiple is important because it allows for the coefficients of the variables to be made the same across the equations, which is necessary for the elimination method to work effectively.

  • What is the least common multiple between 7 and 3, and how does it relate to the equations?

    -The least common multiple between 7 and 3 is 21. This means that to make the coefficients of x the same in both equations, the top equation is multiplied by 3 and the bottom equation by 7.

  • How does the process of multiplying by the LCM affect the original equations?

    -Multiplying by the LCM of the coefficients creates equivalent equations that have the same coefficient for the variable to be eliminated, allowing for the subtraction or addition of the equations to solve for one of the variables.

  • Why does the speaker choose to eliminate the x variable first?

    -The speaker chooses to eliminate the x variable first because it simplifies the process and the numbers involved are easier to work with, leading to a quicker solution for y.

  • What is the significance of the distributive property in this context?

    -The distributive property is significant because it ensures that every term in the equation is multiplied by the LCM multiplier, resulting in equivalent equations that maintain the equality.

  • What is the result of subtracting the two equivalent equations after multiplying by the LCM?

    -Subtracting the two equivalent equations results in an equation with only one variable, which simplifies the process of finding the value of that variable.

  • How does knowing the value of y help in finding the value of x?

    -Knowing the value of y allows for the substitution of this value into one of the original or equivalent equations to solve for x.

  • What is the final solution for the system of equations presented in the transcript?

    -The final solution for the system of equations is x = -1 and y = -1, which is the coordinate point of intersection for the two lines represented by the equations.

  • Why does the speaker prefer to use the original equations to solve for x instead of the equivalent equations?

    -The speaker prefers to use the original equations to solve for x because they have smaller numbers, which are easier to work with and simplify the calculation process.

Outlines

00:00

📘 Solving Systems of Equations Through Multiplication and Elimination

The speaker introduces a method for solving systems of equations by focusing on finding the least common multiple (LCM) of the coefficients to align them for elimination. They explain that while LCMs for 7 and 3 is 21, and for 5 and 4 is 20, the choice of which pairs to align (to eliminate either x or y) is flexible and won't affect the outcome. The speaker opts to eliminate x by multiplying the first equation by 3 and the second by 7, and then shows the process of applying the distributive property to create equivalent equations. They then subtract these equations to solve for y, finding y = -1. Subsequently, using the value of y, they solve for x using one of the original equations, concluding with x = -1. The solution to the system is the point (-1, -1), demonstrating the effectiveness of the method discussed.

Mindmap

Keywords

💡Substitution

Substitution is a method used to solve systems of linear equations where one variable is expressed in terms of another from one equation and then substituted into the other equation. In the script, the instructor mentions substitution as a method that might be initially considered by students but then moves on to discuss more complex scenarios where other methods are necessary.

💡Elimination

The elimination method is a technique for solving systems of linear equations by manipulating the equations so that one variable is eliminated through addition or subtraction, allowing the solution of the other variable. The instructor in the script uses the elimination method by multiplying the equations to align the coefficients of the variables in order to eliminate one of them.

💡Least Common Multiple (LCM)

The least common multiple is the smallest number that is a multiple of two or more integers. In the context of the video, the LCM is used to determine the multipliers needed to make the coefficients of a variable in two equations the same, which is a step in the elimination process. The script mentions finding the LCM of coefficients like 7 and 3, which is 21.

💡Coefficients

Coefficients are the numerical factors that multiply the variables in a linear equation. In the script, the instructor discusses how to manipulate the coefficients of variables to make them the same so that the variables can be eliminated by adding or subtracting the equations.

💡Distributive Property

The distributive property is a fundamental algebraic principle that allows for the multiplication of a term by a sum or difference by multiplying each addend separately and then adding the products. In the script, the instructor emphasizes the importance of applying the distributive property when multiplying each term in the equations by the necessary multipliers to create equivalent equations.

💡Equivalent Equations

Equivalent equations are equations that have the same solutions. In the video, the instructor creates equivalent equations by multiplying the original equations by certain factors to align the coefficients of the variables. These equivalent equations are then used to eliminate variables and find the solution.

💡Multiplying Equations

Multiplying equations involves scaling both sides of an equation by the same factor to maintain equality. The script describes how the instructor multiplies the equations by specific numbers to align the coefficients of the variables, which is a step in the elimination method.

💡Variable Elimination

Variable elimination is the process of removing one variable from the equations to solve for the other. In the script, the instructor eliminates the variable x by subtracting the aligned equations, which have the same coefficient for x, to solve for y.

💡Solving for a Variable

Solving for a variable involves isolating that variable on one side of the equation to find its value. After eliminating x, the instructor in the script solves for y by dividing both sides of the equation by the coefficient of y, which is 43.

💡Coordinate Point of Intersection

The coordinate point of intersection is the solution to a system of equations, represented as an ordered pair (x, y). The script concludes with the instructor finding the values of x and y that satisfy both original equations, which is the point of intersection of the two lines represented by the equations.

Highlights

Introduction to solving systems of linear equations using substitution, elimination, and least common multiples.

Explanation of how to find the least common multiple to align coefficients for elimination.

Multiplying the top equation by 3 and the bottom equation by 7 to eliminate x.

Multiplying the top equation by 4 and the bottom equation by 5 to eliminate y.

The choice between eliminating x or y and its impact on the solution process.

Applying the distributive property to ensure equivalent equations.

Deriving the equivalent equations after multiplication.

Subtracting the equations to eliminate x and solve for y.

Solving for y and finding its value to be negative 1.

Using the value of y to solve for x in the original equations.

Solving for x and finding its value to be negative 1.

Determining the intersection point of the two systems at (-1, -1).

Emphasizing the importance of choosing the method that simplifies calculations.

Highlighting the practicality of using original equations for solving x after finding y.

Conclusion on solving systems of equations by multiplying with two multipliers.

Casual Browsing

Learn to solve a system of equations using substitution

2024-06-12 23:15:00

How to solve linear and quadratic equations and more with Socratic by Google

2024-07-27 05:53:00

Algebra - How To Solve Equations Quickly!

2024-06-13 00:35:00

Learn How To Solve Equations – Understand In 7 Minutes

2024-06-13 00:05:01

How to Solve One-Step Equations | One-Step Equation Steps | Math with Mr. J

2024-06-12 22:10:00

AI Math Solver | Solve any Mathematical Problem in Seconds using AI

2024-07-06 23:55:00