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## Systems of Equations and Inequalities## 4.1 Solving Systems of equations in Two Variables
To solve a system with two equations in two variables we graph them both on
the same axes and
1. Solve the following system of equations graphically.
2. Let's consider all the possible ways that two lines can interact:
It is usually not practical to solve systems by graphing (messy drawings,
points may not be exactly Substitution Method 1. Pick an equation and solve for one of the variables. (Note: It is usually
easiest to solve for a
1. Solve using the substitution method.
2. Solve using the substitution method.
Often the easiest method for solving a system of equations is the
Addition Method 1. If necessary, rewrite each equation in
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 VariablesAn equation of the form 3x - 2y + 4z = 11 is a linear equation in three
variables. The solution
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 SolvingThere are countless applications for systems of equations. We will examine a couple examples
1. How many pints of a 10% salt solution and a 50% salt solution must be
combined to get 44
2. On an algebra test, a total of 75 students had grades of A or B. There
were 5 more students
3. Three brothers invest a total of $35,000 a business. The rst brother earns
3% pro t on his
## 4.4 Solving Systems of Equations Using Matrices
A matrix is a rectangular array of numbers within brackets. The plural of
matrix is
The numbers inside the matrices are referred to as elements. We refer to the
size of the matrix
the corresponding matrix is written:
In order to solve systems of equations using matrices we will attempt to
transform our matrices using
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.
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.
Recall that we call a system of equations inconsistent if there are no common
solution to the Let's examine how to determine, using matrices, if a system is inconsistent or dependent.
1. Solve the system of equations using matrices.
2. Solve the system of equations using matrices.
## 4.5 Determinants and Cramer's Rule
For every 2 The determinant of a 2*2 matrix is denoted and is evaluated as follows:
We will study now a cool rule that gives us a formula for solving systems of
linear equations. We Given the system of equations:
We denote by D the following:
Further we denote with D and Finally, we have Cramer's Rule: For the system of equations Then the solution is given by and
## 4.6 Solving Systems of Linear InequalitiesRecall, we learned how to graph linear inequalities in two variables:
We now investigate how to solve
2. Graph 3. Graph |