# Algebra II: Systems of Equations and Inequalities

On this page we hope to clear up problems you might have with systems of equations and inequalities and their graphs.  Systems of equations are used a lot in other math-related subjects, such as chemistry.  Read on or follow any of the links below to start understanding systems of equations.

Solving systems of equations
3-Variable systems of equations
Systems of inequalities
Quiz on systems of equations and inequalities

## Solving Systems of Equations

Solving systems of equations graphically is one of the easiest ways to solve systems of simple equations (it's usually not very practical for complex equations such as hyperbolas or circles).  However, it is usually covered in elementary algebra (Algebra I) courses.  We have followed this custom on this site.  Follow this link to learn about solving systems of equations.

Another way to solve systems of equations is by substitution.  In this method, you solve an equation for one variable, then you substitute that solution in the other equation, and solve.

Example:

```1.  Problem: Solve the following system:
x + y = 11
3x - y = 5

Solution: Solve the first equation for y (you could
solve for x — it doesn't matter).

y = 11 - x

Now, substitute 11 - x for y in the second
equation.  This gives the equation one variable, which
earlier algebra work has taught you how to solve for.

3x - (11 -x) = 5
3x - 11 + x = 5
4x = 16
x = 4

Now, substitute 4 for x in either equation and
solve for y.  (We use the first equation below.)

4 + y = 11
y = 7

The solution is the ordered pair, (4, 7).
```
The last method, addition, is probably the most complicated, but is necessary when dealing with more complex systems, such as systems with three or more variables.  The idea behind the addition method is to replace an equation with a combination of the equations in the system.  To obtain such a combination, you multiply each equation by a constant and add.  You choose the constants so that the resulting coefficient of one of the variables will be 0.

Example:

```1.  Problem: Solve the following system:
5x + 3y = 7
3x - 5y = -23
Solution: Multiply the second equation by 5 to make the
x-coefficient a multiple of 5.  (This works because
it does not change the equation (see the
multiplication property).)

15x - 25y = -115

Next, multiply the first equation by -3 and add it
to the second equation.  This gets rid of the x
term.

-15x -  9y =  -21
15x - 25y = -115
-----------------
- 34y = -136

Now, solve the second equation for y.  Then substitute
the result into the first equation and solve for x.

-34y = -136
y = 4

5x + 3(4) = 7
5x + 12 = 7
5x = -5
x = -1

The solution is the ordered pair, (-1, 4).
```

## Systems of Equations in 3 Variables

Since you would need a three-dimensional coordinate system to solve systems in three variables, solving graphically is not an option.  Substitution would work, but is usually unmanageable.  Therefore, we will use the addition method, which is basically the same process as it is with systems in two variables.

Example:

```1.  Problem: Solve the following system:
x +  y +  z =  4
x - 2y -  z =  1
2x -  y - 2z = -1
Solution: Start out by multiplying the first equation by -1
and add it to the second equation to eliminate x
from the second equation.

-x  - y - z = -4
x - 2y - z =  1
----------------
-3y - 2z = -3

Now eliminate x from the third equation by
multiplying the first equation by -2 and add it
to the third equation.

-2x - 2y - 2z = -8
2x -  y - 2z = -1
------------------
-3y - 4z = -9

Next, eliminate y from the third equation by
multiplying the second equation by -1 and adding
it to the third equation.

3y +  2z = 3
-3y -  4z = -9
--------------
-2z = -6

Solve the third equation for z.

-2z = -6
z = 3

Substitute 3 for z in the second equation and
solve for y.

-3y - 2z = -3
-3y - 2(3) = -3
-3y - 6 = -3
-3y = 3
y = -1

Lastly, substitute -1 for y and 3 for
z in the first equation and solve for x.

x + (-1) + 3 = 4
x + 2 = 4
x = 2

The answer is (2, -1, 3).
```

## Systems of Inequalities

The easiest way to solve systems of inequalities is to solve them by graphing.  Therefore, it is best if you know how to graph inequalities in two variables (5x - 4y < 13, for example).

Example:

```1.  Problem: Graph y < x.
Solution: First graph the equation y = x.
However, the line must be drawn dashed because the less
than sign tells us the line is not included in the
solution.

Next, test a point that is located above the line and
one that is below the line.  Any point you pick above
the line, such as (0, 2), y is greater
than x, so points above the line are not included
in the solution.  Points below the line, such as (3, -3)
have a y-value that is less than the x-value,
so all points below the line are included in the
solution.  Example Graph.
```
To solve a system or conjunction of inequalities, it is easiest to graph each of the inequalities and then find their intersection.

Example:

```1.  Problem: Graph the following system:
2x + y >= 2
4x + 3y <= 12
(1/2) <= x <= 2
y >= 0
Solution: Example Figure.
```

Take the quiz on systems of equations and inequalities.  The quiz is very useful for either review or to see if you've really got the topic down.

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