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Inequalities
Using MATLAB to Solve Linear Inequalities
Equations and Inequalities
Graph Linear Inequalities in Two Variables
Solving Equations & Inequalities
Teaching Inequalities:A Hypothetical Classroom Case
Graphing Linear Inequalities and Systems of Inequalities
Inequalities and Applications
Solving Inequalities
Quadratic Inequalities
Inequalities
Solving Systems of Linear Equations by Graphing
Systems of Equations and Inequalities
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Inequalities
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Solving Inequalities

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Systems of Equations and Inequalities

4.1 Solving Systems of equations in Two Variables

De nition 1. A system of linear equations is two or more linear equations to which we try to
find a common solution.

EX 1. Consider the equations:

A solution to a system of linear equtions is an ordered pair or pairs that satisfy ALL equations
in the system. Let's verify that (3, 7) is a solution to the given system.

4.1.1 Solve Systems of Linear Equations Graphically

To solve a system with two equations in two variables we graph them both on the same axes and
common solution is the point where they intersect.

EX 2. -

1. Solve the following system of equations graphically.

2. Let's consider all the possible ways that two lines can interact:

4.1.2 Solving Systems by Substitution

It is usually not practical to solve systems by graphing (messy drawings, points may not be exactly
clear) so we need more precies methods which we now develop. Our rst algebraic method is called
substitution.

Substitution Method

1. Pick an equation and solve for one of the variables. (Note: It is usually easiest to solve for a
variable with coefficient of 1.)
2. Substitute the expression found for the variable in step 1 into the other equation. We
now have an equation with only one variable.
3. Solve the equation in step 2 to find the value of one of the variables.
4. Substitute the value found in step 3 into the equation from step 1. Solve the equation to find
the other variable.
5. Check your solution in all of the equations in the system.

EX 3. -

1. Solve using the substitution method.

2. Solve using the substitution method.

4.1.3 Solving Systems Using the Addition Method

Often the easiest method for solving a system of equations is the addition method (also known as
the elimination method). Before we list the steps, let's look at an example.

EX 4. Solve the following system of equations using the addition method.

Addition Method

1. If necessary, rewrite each equation in standard form.
2. If necessary, multiply one or both equations by a constant so that when the equations are
added, the sum will contain only one variable.
3. Add the equations. We get an equation in one variable.
4. Solve the equation found in step 3.
5. Substitute the value found in step 4 into either of the original equations and solve for the other
variable.
6. Check your solution in all of the equations in the system.

EX 5. -

1. Solve using the addition method:

2. Solve using the addition method:

3. Solve using the addition method:

4. Solve using the addition method:

4.2 Solving Systems of Linear Equations in Three Variables

An equation of the form 3x - 2y + 4z = 11 is a linear equation in three variables. The solution
to such an equation is an ordered triple of the form (x, y, z). For, example a solution of the above
equation is (1, 2, 3). We will consider systems of three equations in three variables.
To solve systems of equations in three variables we will use the methods we have already intro-
duced, namely the substitution and addition methods.

EX 6. -

1. Solve the following system using the substitution method.

2. Solve the following system using the addition method.

3. Solve the following system using the addition method.

4.3 Applications and Problem Solving

There are countless applications for systems of equations. We will examine a couple examples

EX 7. -

1. How many pints of a 10% salt solution and a 50% salt solution must be combined to get 44
pints of a 40% salt solution?

2. On an algebra test, a total of 75 students had grades of A or B. There were 5 more students
with A's than B's. Find the number of students who had an A and the number of students
who had a B.

3. Three brothers invest a total of $35,000 a business. The rst brother earns 3% pro t on his
investment. The second brother earns a 5% pro t on his investment. The third brother earns
7% pro t on his investment. The third brother invested $5000 more than the sum of the other
two brothers. After one year, the pro t earned was $2050. How much did each brother invest?

4.4 Solving Systems of Equations Using Matrices

4.4.1 Matrices

A matrix is a rectangular array of numbers within brackets. The plural of matrix is matrices.
EX 8. Here are a couple matrices:

The numbers inside the matrices are referred to as elements. We refer to the size of the matrix
by the number of rows and columns, thus the left matrix has 2 rows and 3 columns so it is a 2*3
matrix. A square matrix is a matrix which has the same number of rows as columns.
We will use matrices to solve our systems of equations. In order to do this we will represent our
system with an augmented matrix. Consider the system of equations

the corresponding matrix is written:

EX 9. Express the following system of linear equations with an augmented matrix.

4.4.2 Solve Systems of Linear Equations

In order to solve systems of equations using matrices we will attempt to transform our matrices using
3 simple transformations. Our goal is to rewrite the matrix in the following form (called triangular
form):

To transform a matrix into this form we have the following row transformations:

Procedures for Row Transformations

1. Any row may be multiplied (or divided) by a nonzero number.
2. All the numbers in a row may be multiplied by any nonzero real number, then added to any other row.
3. The order of the rows may be switched.

EX 10. -

1. Solve the following system of equations using matrices.

2. Solve the following system of equations using matrices.

3. Solve the following system of equations using matrices.

4.4.3 Recognize Inconsistent and Dependent Systems

Recall that we call a system of equations inconsistent if there are no common solution to the
equations of the system (parallel lines). A dependent system is one with an in nite number of
solutions.

Let's examine how to determine, using matrices, if a system is inconsistent or dependent.

EX 11. -

1. Solve the system of equations using matrices.

2. Solve the system of equations using matrices.

4.5 Determinants and Cramer's Rule

4.5.1 Evaluate a Determinant of a 2*2 Matrix

For every 2*2 matrix we can find a number associated with it called the determinant.

The determinant of a 2*2 matrix is denoted and is evaluated as follows:

EX 12. Evaluate the following determinants:

4.5.2 Cramer's Rule

We will study now a cool rule that gives us a formula for solving systems of linear equations. We
will only study this for systems with two variables but it works for three or more variable system of
equations. First some notation.

Given the system of equations:

We denote by D the following:

Further we denote with D_x and D_y:

and

Finally, we have Cramer's Rule: For the system of equations

Then the solution is given by

and

EX 13. Solve the system of equations using Cramer's rule.

REMARK 1. When D = 0:
If D = 0, D_x = 0, and D_y = 0, then the system is dependent.
If D = 0 and either D_x ≠ 0 or D_y ≠ 0, then the system is inconsistent.

4.6 Solving Systems of Linear Inequalities

Recall, we learned how to graph linear inequalities in two variables:

EX 14. Graph the inequality 2x + 3y > 6

We now investigate how to solve systems of linear inequalities graphically.
To do this, we graph each inequality on the same axes. The solution is the set of points whose
coordinates satisfy all the inequalities in the system.

EX 15. -
1. Graph:

2. Graph

3. Graph