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)

Example: Substitution

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

Solve for x, y, and z

(1)(2)(3)

Example: Substitution

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

Solve for x, y, and z

(1)(2)(3)

Example: Substitution

Replace equation (3) with the new equation

Solve for x, y, and z

(1)(2)(3)

Example: Substitution

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

Solve for x, y, and z

(1)(2)(3)

Example: Substitution

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

Solve for x, y, and z

(1)(2)(3)

Example: Substitution

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

Solve for x, y, and z

(1)(2)(3)

Example: Substitution

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

Solve for x, y, and z

(1)(2)(3)

Example: Substitution

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

Solve for x, y, and z

(1)(2)(3)

Example: Substitution

Now we’ve successfully solved a system of equations

Solve for x, y, and z

(1)(2)(3)

Example: EliminationSolve for x, y, and z

(1)(2)(3)

Example: EliminationSolve for x, y, and z

(1)(2)(3)

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

Example: EliminationSolve for x, y, and z

(1)(2)(3)

Start by multiplying equation (2) by -1.

(1)-(2)

Example: EliminationSolve for x, y, and z

(1)(2)(3)

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

(1)-(2)

(1) - (2)

Example: EliminationSolve for x, y, and z

(1)(2)(3)

Replace equation (1) with this new equation.

Example: EliminationSolve for x, y, and z

(1)(2)(3)

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

(1)-(3)

Example: EliminationSolve for x, y, and z

(1)(2)(3)

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

(1)-(3)

(1) – (3)

Example: EliminationSolve for x, y, and z

(1)(2)(3)

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

Example: EliminationSolve for x, y, and z

(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

Example: MatricesSolve for x, y, and z

(1)(2)(3)

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

x solution

Example: MatricesSolve for x, y, and z

(1)(2)(3)

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

y solution

Example: MatricesSolve for x, y, and z

(1)(2)(3)

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

z solution

Example: MatricesSolve for x, y, and z

(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)

Example: MatricesSolve for x, y, and z

(1)(2)(3)

The second entry in each row will be the y coefficient

x(1)(2)(3)

y

Example: MatricesSolve for x, y, and z

(1)(2)(3)

The third entry in each row will be the z coefficient

x(1)(2)(3)

y z

Example: MatricesSolve for x, y, and z

(1)(2)(3)

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

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

RHS

Example: MatricesSolve for x, y, and z

(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)

Example: MatricesSolve for x, y, and z

(1)(2)(3)

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

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)

Example: MatricesSolve for x, y, and z

(1)(2)(3)

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

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)

Example: MatricesSolve for x, y, and z

(1)(2)(3)

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

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

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)

Example: MatricesSolve for x, y, and z

(1)(2)(3)

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

(1)(2)(3)

(1)(2)(3)

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

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

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

Example: MatricesSolve for x, y, and z

(1)(2)(3)

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

(1)(2)(3)

(1)(2)(3)

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

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

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

Example: MatricesSolve for x, y, and z

(1)(2)(3)

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

(1)(2)(3)

(1)(2)(3)

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

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

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

Example: MatricesSolve for x, y, and z

(1)(2)(3)

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

(1)(2)(3)

(1)(2)(3)

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

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

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

Example: MatricesSolve for x, y, and z

(1)(2)(3)

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

(1)(2)(3)

(1)(2)(3)

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

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

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

Example: MatricesSolve for x, y, and z

(1)(2)(3)

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

(1)(2)(3)

(1)(2)(3)

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

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

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

Example: MatricesSolve for x, y, and z

(1)(2)(3)

x = 4y = 3z = 2

(1)(2)(3)

x y z RHS