1. Consider the two equations [ 4x + 22 8x + 10 , 3x + 18 = 5x +8 ]
2. What do you notice about these equations?
3. What does it mean to have one solution?
4. Plug a few points into the original equations and check to show that the system of
equations has one solution.
1. The two equations are in the form of linear equations with variables x.
2. I notice that the coefficients of x in the first equation are not the same as the coefficients of x in the second equation. This indicates that the system of equations is consistent and has a unique solution.
3. Having one solution means that there is one unique value for the variable (in this case, x) that satisfies both equations simultaneously.
4. Let's plug in some values for x into the original equations:
For the first equation:
When x = 1, 4(1) + 22 = 26, 8(1) + 10 = 18
So, the first equation becomes: 26 = 18, which is false.
For the second equation:
When x = 1, 3(1) + 18 = 21, 5(1) + 8 = 13
So, the second equation becomes: 21 = 13, which is false.
Therefore, the system of equations does not have a solution.
Now try writing and solving your own systems of linear equations: one with infinite
solutions, one with no solution, and one with one solution. Write a real-world scenario
that goes with at least one system of equations. Show your work
1. System with Infinite Solutions:
Equation 1: 2x + 3y = 12
Equation 2: 4x + 6y = 24
In this system, both equations represent the same line.
To solve:
Dividing the second equation by 2, we get:
2x + 3y = 12, which is the same as Equation 1.
Therefore, the system has infinitely many solutions.
Real-world scenario:
A company manufactures two types of products, where the second product is exactly double the first product in terms of cost and revenue.
2. System with No Solution:
Equation 1: 3x + 2y = 10
Equation 2: 3x + 2y = 12
In this system, the lines represented by both equations are parallel and will never intersect.
To solve:
Subtracting Equation 1 from Equation 2:
12 - 10 = 2, which is a contradiction.
Therefore, the system has no solution.
Real-world scenario:
In a sales scenario, one company offers a product for $10 and a competitor offers a similar product for $12, but both have the same features and quality.
3. System with One Solution:
Equation 1: 5x + 2y = 11
Equation 2: 3x - y = 4
In this system, the equations form two intersecting lines with one unique solution.
To solve:
Solving Equation 2 for y:
y = 3x - 4
Substitute back into Equation 1:
5x + 2(3x - 4) = 11
5x + 6x - 8 = 11
11x - 8 = 11
11x = 19
x = 19/11
Substitute x back into y = 3x - 4:
y = 3(19/11) - 4
y = 57/11 - 44/11
y = 13/11
Therefore, the system has one solution at (19/11, 13/11).
Real-world scenario:
A bakery sells cookies and muffins, where the total revenue from selling 5 cookies and 2 muffins is $11, and the total cost revenue from selling 3 cookies and y muffins is $4. The unique solution represents the point where the bakery maximizes its profit.