Reviewing Systems of Equations

Reviewing Systems of Equations• Substitution• Elimination• Matrices

Some Important Points• Several (2+) variables need to be solved for• To find an exact solution, you need to

generate the same number of equations as you have variables

• For example, you’ll need 3 equations to work with if you’re going to try and solve for 3 variables

Three Methods for Solving

• Substitution – substituting variables to form equations with single variables

• Elimination – combining equations to eliminate variables (convenient only in certain situations)

• Matrices – use of matrices and their operations to arrive at a solution

Example: SubstitutionSolve for x, y, and z

(1)(2)(3)

Example: Substitution

Transform equation (2) to have z in terms of x.

Solve for x, y, and z

(1)(2)(3)


Replace the original equation (2) with the new equation.


(1)(2)(3)


Use the same technique on equation (3) to get y in terms of x


(1)(2)(3)


Replace equation (3) with the new equation


(1)(2)(3)


Now substitute the expression for z in terms of x from equation (2) into equation (1).


(1)(2)(3)


Continue to modify this equation by substituting equation (3) into equation (1).


(1)(2)(3)


You’ll notice that this new equation has only 1 variable (x) in it. We can readily solve for x and use this result in further calculations


(1)(2)(3)


Now, we can use the result of x=4 in equation (3) to solve for y.


(1)(2)(3)


Finally, we’ll use the result of x=4 in equation (2) to determine the value for z.


(1)(2)(3)


Now we’ve successfully solved a system of equations


(1)(2)(3)

Example: EliminationSolve for x, y, and z

(1)(2)(3)


(1)(2)(3)

We’ll use elimination to verify our work from before. We can start be subtracting equation (2) from equation (1)


(1)(2)(3)

Start by multiplying equation (2) by -1.

(1)-(2)


(1)(2)(3)

Add equation (1) and equation –(2). Notice that the z term is eliminated.

(1)-(2)

(1) - (2)


(1)(2)(3)

Replace equation (1) with this new equation.


(1)(2)(3)

Now subtract equation (3) from equation (1) using a similar approach, start with multiplying equation (3) by -1

(1)-(3)


(1)(2)(3)

Add equation (1) and equation –(3). Notice that the x terms cancel.

(1)-(3)

(1) – (3)


(1)(2)(3)

Use the result of y=3 in equation (1) to determine the value of x.


(1)(2)(3)

Use the result of x=4 in equation (2) to find z.

Example: MatricesSolve for x, y, and z

(1)(2)(3)

The first step is to set up a matrix describing this system of equations. This matrix will be manipulated so that there is a diagonal of 1’s and corresponding values on the other side of the bar

(1)(2)(3)

x y z RHS


(1)(2)(3)

In this form, the top row will contain the solution for x

x solution


(1)(2)(3)

In this form, the middle row will contain the solution for y

y solution


(1)(2)(3)

In this form, the bottom row will contain the solution for z

z solution


(1)(2)(3)

The first step is to set up a matrix describing this system of equations. To do this, each equation will be transcribed into a row. These 3 rows will be stacked on top of each other to form the matrix. The first entry in a row will be the x coefficient.

x(1)(2)(3)


(1)(2)(3)

The second entry in each row will be the y coefficient

x(1)(2)(3)

y


(1)(2)(3)

The third entry in each row will be the z coefficient

x(1)(2)(3)

y z


(1)(2)(3)

The final entry is the right hand side (RHS) of each equation.

x y z(1)(2)(3)

RHS


(1)(2)(3)

For matrix operations, we are allowed to do the following things in succession:

1. We can swap rows at any time so long as all entries retain their position within a row

(1)(2)(3)

Swap (1) and (2)

(1)(2)(3)


(1)(2)(3)


2. We can multiply any row by a constant so long as it’s applied to each entry within the row

Multiply (3) by -1

(1)(2)(3)

(1)(2)(3)


(1)(2)(3)


3. We add equations together and replace one of those equations with the results of the addition

Replace (2) with (3)+(2)

(1)(2)(3)

(1)(2)(3)


(1)(2)(3)



3. We add equations together and replace one of those equations with the results of the additionReplace (2) with

3(2)+(3)(1)(2)(3)

(1)(2)(3)


(1)(2)(3)

Replace (3) with 3(1)+(3)

(1)(2)(3)

(1)(2)(3)





(1)(2)(3)


(1)(2)(3)

(1)(2)(3)





(1)(2)(3)

Replace (3) with -1/9(3)

(1)(2)(3)

(1)(2)(3)





(1)(2)(3)


(1)(2)(3)

(1)(2)(3)





(1)(2)(3)

Replace (2) with (2)+6(3)

(1)(2)(3)

(1)(2)(3)





(1)(2)(3)

x = 4y = 3z = 2

(1)(2)(3)

x y z RHS

Reviewing Systems of Equations

Documents

Transcript of Reviewing Systems of Equations