7.1.3 Solving a System of Equations Using Elimination
TLDRIn this informative video, Mr. Banker demonstrates how to solve systems of equations using the elimination method. He explains the process step by step, starting with an example where x and y values are found by eliminating one variable. He then showcases a scenario with no solutions, where the resulting equation is nonsensical, indicating parallel lines. Lastly, he illustrates the concept of infinitely many solutions, where the equations represent overlapping lines, and any point on these lines is a solution. The video is a comprehensive guide to understanding different outcomes in solving systems of equations.
Takeaways
- 📚 The method of solving a system of equations by elimination involves adding two equations to eliminate one variable.
- 🤔 To effectively use elimination, the coefficients of one variable must be opposites or have a least common multiple that allows them to cancel each other out.
- 🔢 Example: To eliminate x, the coefficients 2 and 3 are multiplied by a least common multiple (LCM) of 6, resulting in 6x and -6x when combined.
- 🎯 When the coefficients are properly adjusted, the addition of the equations leads to the cancellation of the targeted variable, leaving the other variable alone.
- 🔍 After finding the value of one variable, it should be substituted back into one of the original equations to solve for the other variable.
- 🌟 The result is expressed as an ordered pair (x, y) representing the solution to the system of equations.
- ⚠️ A no solution case occurs when the equations result in a mathematical statement like 0 = 8, which is impossible, indicating the lines are parallel and never intersect.
- 🌀 Infinitely many solutions arise when the adjusted equations result in a true statement like 0 = 0, meaning the lines overlap and every point on the line is a solution.
- 📊 Graphical interpretation: If the mathematical process results in a 0 = 0 statement, it corresponds to the lines being coincident or overlapping on a graph.
- 👍 The video provides a comprehensive guide to understanding the elimination method, including scenarios with a unique solution, no solution, and infinitely many solutions.
Q & A
What is the main method discussed in the video for solving a system of equations?
-The main method discussed in the video is the elimination method.
What is the goal of the elimination method?
-The goal of the elimination method is to add the two equations together in such a way that one of the variables is eliminated, making it easier to solve for the remaining variable.
How does the video demonstrate finding opposite coefficients for the elimination method?
-The video demonstrates finding opposite coefficients by looking for a least common multiple between the coefficients of the variable you want to eliminate, in this case, the x's.
What is the first example's system of equations and how is it solved?
-The first example's system of equations is 2x + 3y = 5 and 3x + 5y = 21. It is solved by multiplying the first equation by 3 and the second equation by 2 to make the coefficients of x opposites, then adding them to eliminate x, resulting in a new equation to solve for y, and subsequently using the found y value to find the x value.
What does the video mean by a 'no solutions' case?
-A 'no solutions' case occurs when the equations after attempting to eliminate a variable result in a mathematical statement that doesn't make sense, such as 0 = 8. This indicates that the lines represented by the equations are parallel and will never intersect.
How does the video explain the concept of 'infinitely many solutions'?
-The video explains 'infinitely many solutions' by showing that when the coefficients of the variables are made to cancel each other out and the resulting equation is true (0 = 0), it means that every point on the line represented by the equation is a solution, hence infinitely many solutions exist.
What is the second example's system of equations and what is the conclusion?
-The second example's system of equations is x - 3y = -2 and 2x - 6y = 4. After attempting to eliminate x, the result is a true statement (0 = 0), leading to the conclusion that there are infinitely many solutions.
How does the video relate the 'no solutions' and 'infinitely many solutions' cases to their graphical representation?
-The video relates the 'no solutions' case to the lines being parallel and never intersecting, while the 'infinitely many solutions' case is related to the lines overlapping each other, intersecting everywhere along the line.
What is the third example's system of equations and how does it lead to infinitely many solutions?
-The third example's system of equations is 4x - 5y = 2 and -12x + 15y = -6. By multiplying the first equation by 3 to make the coefficients of x opposites and adding them, the result is a true statement (0 = 0), indicating infinitely many solutions.
How does the video suggest rearranging the equation to find a solution in the 'infinitely many solutions' case?
-The video suggests rearranging one of the equations to the form y = mx + b, where m is the slope and b is the y-intercept. In this case, it would be y = 4/5x - 2/5, meaning any point on this line is a solution to the system of equations.
What is the final solution in the first example as an ordered pair?
-The final solution in the first example is (x, y) = (-2, 3).
Outlines
📚 Introduction to Solving Systems of Equations by Elimination
This paragraph introduces the concept of solving systems of equations using the elimination method. The speaker, Mr. Banker, explains that the goal is to eliminate one variable by adding the two equations together. He emphasizes the need for opposite coefficients for the variable being eliminated and demonstrates the process using an example. The speaker walks through the steps of finding a least common multiple for the coefficients, multiplying the equations accordingly, and adding them to cancel out the variable. The result is a simpler equation to solve for the remaining variable. The process is illustrated with an example that leads to the solution, x = -2 and y = 3, presented as an ordered pair.
🔍 Exploring No Solutions and Infinitely Many Solutions Cases
In this paragraph, the speaker continues the discussion on solving systems of equations, focusing on special cases where there are no solutions or infinitely many solutions. The no solution case is demonstrated using a set of equations that, when attempted to be solved by elimination, results in a contradiction (0 = 8), indicating that the lines represented by the equations are parallel and never intersect. The infinitely many solutions scenario is explained using another set of equations. The process shows that the equations are equivalent, leading to the conclusion that any point on the line formed by the equation y = 4/5x - 2/5 is a solution to the system. This is visually depicted as the lines overlapping each other, intersecting at every point along their length.
Mindmap
Keywords
💡System of Equations
💡Elimination Method
💡Coefficients
💡Least Common Multiple (LCM)
💡No Solutions
💡Infinite Solutions
💡Graphing
💡Variables
💡Ordered Pair
💡Substitution
💡Parallel Lines
Highlights
The video discusses solving systems of equations using the elimination method.
In elimination, the goal is to add two equations together to eliminate one variable.
To eliminate a variable, opposite coefficients are needed in front of that variable.
The process involves finding the least common multiple of the coefficients.
An example is given where x is eliminated by multiplying the first equation by 3.
After eliminating x, the resulting equation is 19y = 57, leading to y = 3.
The x value is then found by substituting y back into one of the original equations, resulting in x = -2.
The solution is presented as an ordered pair (x = -2, y = 3).
A no solution case is demonstrated where the resulting equation is 0 = 8, indicating no intersection.
In the no solution case, the lines are parallel and never intersect.
Another example is shown where the system has infinitely many solutions due to the lines overlapping.
The infinite solutions case is represented by the equation 0 = 0, a true mathematical statement.
Graphically, the lines intersect everywhere, indicating any point on the line is a solution.
The video concludes by expressing gratitude to the viewers for watching.
The method for solving systems of equations is applicable in various mathematical and real-world contexts.
The video provides a clear and detailed explanation of the elimination method, suitable for educational purposes.
Transcripts
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