The provided text is a transcript of a video lesson on solving systems of linear equations with two unknowns using the substitution method. The instructor presents three examples to illustrate the method.
Example 1:
The problem involves a parking lot with 12 vehicles, consisting of cars and motorcycles. The difference between the number of cars and double the number of motorcycles is 3. The instructor sets up a system of two linear equations with two unknowns (x for cars and y for motorcycles) and solves it using the substitution method. The solution is x = 9 and y = 3, meaning there are 9 cars and 3 motorcycles in the parking lot.
Example 2:
The instructor presents a system of two linear equations with two unknowns (x and y) and asks the viewer to solve it using the substitution method. The instructor then provides the solution, which is x = -1 and y = 4.
Example 3:
The instructor presents a system of two linear equations with two unknowns (x and y), but the equations are not in the standard form. The instructor first transforms the equations into the standard form by multiplying both sides of the equations by the least common multiple (LCM) of the denominators. The instructor then solves the system using the substitution method. The solution is x = 1 and y = 4.
Throughout the lesson, the instructor emphasizes the importance of choosing the simpler equation to isolate one of the unknowns and substituting that value into the other equation to solve for the other unknown.
Here are the key facts extracted from the text:
1. The problem involves a system of two equations with two unknowns.
2. The system of equations is to be solved using the substitution method.
3. The first equation is x + y = 12.
4. The second equation is x - 2y = 3.
5. The solution to the system of equations is x = 9 and y = 3.
6. The values of x and y must satisfy both equations.
7. The solution to the system of equations is an ordered pair (x, y) = (9, 3).
8. Another system of equations is 5x + y = -1 and 3x + 4y = 13.
9. The solution to this system of equations is x = -1 and y = 4.
10. The values of x and y must satisfy both equations.
11. The solution to the system of equations is an ordered pair (x, y) = (-1, 4).
12. A third system of equations is x/3 + y = 4 and 2x + y/2 = 1.
13. The system of equations can be transformed into an equivalent system by multiplying both sides of the second equation by 2.
14. The resulting system of equations is 3x + 4y = 19 and 2x + y = 6.
15. The solution to this system of equations is x = 1 and y = 4.
16. The values of x and y must satisfy both equations.
17. The solution to the system of equations is an ordered pair (x, y) = (1, 4).