9.1 Determining number of solutions to linear equations
Now that we have learned about linear equations, linear relations, linear function in the previous chapters, we are to continue learning more about these, by solving linear systems.
Linear systems as the word suggests, is a set of linear equations that would intersect at some point, say for example 2xy= 9 and x + 3y = 6, we know from solving them by either method that they would meet at (3,3).
In section 1 of this chapter, we will practice more on determining how many solutions there are in each of the systems given to us. The solution of the linear systems is basically where the two lines would intersect. When there are no solutions, then this means that the lines are parallel. There are also some instances when there are infinite solutions which mean that the two lines lie on top of each other.
Solving linear systems can come in three methods. We can use the graphing method, the elimination method, and the substitution method. In this chapter we will get to focus on these different methods and at the same time learn the different applications of solving linear systems.
In the graphing method, we just simply graph these two equations and see all the values of x and y and see where they intersect, if they intersect at all.
In the elimination method, we can use addition or subtraction to eliminate one variable to solve for the other. After solving for the value of the variable, we would substitute that value to one of the equation to get the value of the other variable and to find where these lines would intersect.
For the substitution method, we pick one of the equations from the system and use that to solve for either x or y. After that we substitute the value of the chosen variable to the other equation to solve for the variable we are solving for.
In the last parts of these chapters, we will learn all about the different applications of linear equations like solving money problems, distance problems, unknown number problems and rectangle problems.
Determining number of solutions to linear equations
Lessons
Notes:
One Solution: Two linear equations intersect at one point. It is a consistent system of independent equation.
No Solution:Two linear equations are parallel to each other. It is an inconsistent system.
Infinite Solution:

a)
3x + y = 7
4x + y = 7 
b)
6x + 2y = 10
3x + y = 5 
c)
x  y = 3
3x  3y = 6


a)
one solution

b)
no solutions
