Solving Systems of Two Equations and Two Unknowns: Graphing, Substitution, and Elimination
TLDRThe video explains various techniques for solving systems of linear equations, including graphing the equations to find the point where the lines intersect, substitution by solving one variable in terms of the other, and elimination by adding or subtracting equations to cancel out a variable. These techniques are applied to real-world examples like determining how many hours Penelope and Artemis worked given facts about their total hours worked and the relationship between their individual hours.
Takeaways
- 😀 Systems of linear equations often have one solution that satisfies both equations, represented by the point where the lines intersect on a graph.
- 😕 Parallel lines have no solution since they never intersect, while overlapping lines have infinitely many solutions.
- 📏 Techniques like substitution and elimination can be used to algebraically solve for the solution without graphing.
- 🔢 Substitution involves solving for one variable and substituting it into the other equation.
- 🔢 Elimination involves adding or subtracting equations so that one variable cancels out.
- ✏️ Real world problems can be modeled as systems of equations and solved algebraically.
- 👥 The script models a workplace example with total hours worked and relative hours between two people.
- 📈 Setting up and solving the system gives the actual hours worked by each person.
- 👍 Checking the solution by plugging back into the original equations is a good validation step.
- 🧮 These techniques allow solving more complex systems that may not be easily solvable by guessing.
Q & A
What are the two main methods described in the video for solving a system of linear equations?
-The two main methods are substitution, where you solve for one variable and substitute it into the other equation, and elimination, where you manipulate the equations to eliminate one variable.
When graphing two linear equations, what does it mean if the lines are parallel?
-If the lines are parallel they will never intersect. This means there is no solution to the system of equations.
What does it mean if two linear equations describe the same line when graphed?
-If two linear equations describe the same line, then there are infinitely many solutions. Every point on the line satisfies both equations.
In the example with Penelope and Artemis, why is substitution an easy method?
-In the example, one equation already has P solved in terms of A. This makes substitution very straightforward - just substitute the expression for P into the other equation.
What is the least common multiple method mentioned when discussing the elimination method?
-Sometimes when adding/subtracting two equations, the variables do not cancel out cleanly. In this case, the equations can be multiplied by constants so the variables have common coefficients that allow cancellation.
What are some reasons the professor gives for why systems of equations are useful?
-He mentions they allow solving more complex problems than guess-and-check, and allow solving real-world problems like the workplace example given.
What happens if two linear equations have no solution?
-If two linear equations graphed as lines have no intersection point, there is no solution that satisfies both equations. On a graph, this happens if the lines are parallel.
What is significant about the point where two linear equation graphs intersect?
-The intersection point satisfies both equations, since it lies on both lines. So its coordinates give values for x and y that solve the system.
Why can't you generally use guess-and-check for systems of equations?
-For simple systems, guess-and-check could work. But the professor says for more complex systems, the algebra techniques allow solving problems guess-and-check can't handle.
What happens if two linear equations actually represent the same line?
-If two equations graph as the exact same line, then every point on the line solves both equations. So there are infinitely many solutions in this case.
Outlines
😊 Solving Systems of Linear Equations
This paragraph discusses different methods for solving systems of two linear equations with two unknowns. It first shows graphically how the solution is the point where the lines representing the equations intersect. It then explains the substitution method, where you solve one variable in terms of the other and substitute it into the second equation. Next is the elimination method, where you manipulate the equations to cancel out one variable and solve for the other one. It notes that generally there is one solution where the lines intersect once, no solutions if the lines are parallel, and infinite solutions if the lines overlap. It concludes by stating why solving systems of equations is useful.
😃 Applying Systems to a Word Problem
This paragraph shows how to apply systems of linear equations to solve a real-world word problem. Two people worked a total of 70 hours, with one working 2 hours less than twice as much as the other. This translates to a system of two equations with two unknowns. The substitution method provides the solution that one person worked 24 hours and the other worked 46 hours. It emphasizes that while this problem could be solved by guess-and-check, using algebra allows solving more complex problems where guessing is not feasible.
Mindmap
Keywords
💡linear systems
💡substitution
💡elimination
💡graphing
💡real world examples
💡solutions
💡lines
💡variables
💡equations
💡unknowns
Highlights
Graphing the equations to find where the lines intersect visually shows the solution to the system.
The substitution method involves solving one variable in terms of the other and substituting it into the second equation.
The elimination method involves combining equations so one variable disappears, allowing you to solve for the other variable.
If two lines intersect once, there is one unique solution to the system.
If two lines are parallel, there is no solution to the system.
If two lines represent the same equation, there are infinitely many solutions.
Real world situations like number of hours worked can be modeled as a system of equations.
Write one equation representing the total of some quantity, and another equation representing a relationship between unknowns.
Substitution is an easy method when one equation already has a variable solved for.
Always check your solution by plugging the values back into the original equations.
Systems of equations allow solving problems algebraically that would be difficult by guessing.
The point where two lines intersect represents values that satisfy both equations.
Adding or subtracting equations can eliminate a variable and allow solving for the other one.
Multiplying equations by constants can allow you to combine them to eliminate a variable.
If graphs are messy or unavailable, algebraic methods are essential for solving systems.
Transcripts
Browse More Related Video
Solving Systems of Equations By Elimination & Substitution With 2 Variables
Simultaneous Equations - Tons of Examples!
7.1.3 Solving a System of Equations Using Elimination
Ch. 10.1 Systems of Linear Equations in Two Variables
SOLVING SYSYEM OF NONLINEAR EQUATIONS || PRECALCULUS
Graphing Systems of Linear Inequalities
5.0 / 5 (0 votes)
Thanks for rating: