3.a Linear System of Equations Flashcards
(20 cards)
What are the two common methods used to solve a system of linear equation?
1) Substitution method
2) Combination method
What is the “substitution” method?
Isolate one of the variables, and then insert that equation into the corresponding variable in the other equation.
This will give you an equation with only 1 unknown variable.
With these equations, which is best to isolate?
4y = 5 + b
and
6b = 12y + 6
4y = 5 + b
(can do any but just picking this)
Its best to isolate for b,
b = 4y - 5
as isolating for y gives..
y = 5/4 + b/4 = 5 + b / 4
–> just more difficult to work with
What is the “combination” method?
In a system of equation, we can add one equation to another equation (or subtract them) in order to eliminate one variable and solve for the other variable.
In this equation:
4x + 3y = 12
what is the coefficient of y?
3
In this equation:
x + y = 12
what is the coefficient of x?
1
Since x = 1x
What is the “coefficient”?
The coefficient is the number that multiplies a variable
If a variable has the same or opposite coefficient in two equations, we can use the __________
Combination method!
as 6x in one, and -6x, we can ADD the equations to eliminate the x terms
Similarly, with 6x and 6x we can SUBTRACT the equations to eliminate the x terms
What can we do if the variables in the two equations have different coefficients?
We can multiply them by the LCM and then use the combination method!
How can we solve this?
2x + y = 3
6x + 9y = 12
2x + y = 3
(multiply this by x3)
6x + 3y = 9
6x + 9y = 12
= 6y = 3
When should you use the “subtitution” method?
When one of the equations can easily be manipulated to isolate one of the variables on one side of the equation
When should you use the “combination” method”?
When neither equation can easily be manipulated to isolate one of the variables
How many solutions can a system of linear equations have?
ZERO, ONE, or infinitely many..
When does a linear equation have ZERO solutions?
If a system of linear equations is equivalent to one in which the variable coefficients are equal, but constant terms are not, then that linear system will have NO solution.
eg. 0 = 4
Hence, no solution
What is the solution to these linear equations?
2x + 3y = 12
2x + 3y = 8
Since 0 = 4 is obviously impossible, these linear equations have NO solution
If a is constant, and the system has no solution, what is the value of a?
1
As, 1x + 3y = 1
(x2)
2x + 3y = 2
2x + 3y = 3
0 = 1
Makes no sense, ZERO solutions
How can a system of linear equations have infinitely many solutions?
If the two equations are identical
How many solutions are there:
3x - 2y = 8
3x - 2y = 8
The two equations lie on top of one another, they are IDENTICAL
So there are infinitely many solutions
If you solve a system of equations and the outcome is 0=0, how many solutions are there?
Infinitely many
If you solve a system of equations and the outcome is 0=k, how many solutions are there?
(where k is nonzero)
ZERO solutions
