Section 4.1 Solving Systems of Linear Equations by Graphing
A system of linear equations is a set of two or more linear equations that are to be
solved at the same time.
The solution to a system of linear equations in two variables is an ordered pair that
satisfies all equations in the system – that is, when the ordered pair coordinates are
substituted into all equations, the resulting numeric equations are all true.
We can solve a system of linear equations graphically.
To Solve a System of Two Linear Equations Graphically:
1. Graph both equations on the same set of axes.
2. If the lines intersect, the point of intersection is the solution.
3. If the lines are parallel there is no solution.
4. If the lines coincide then the solutions are infinite and they are all of the points on the
line.
Further Descriptors
If the system of equations has at least one ordered pair solution (which means that the
graphs either intersect or coincide), we say that the system is consistent.
If the system of equations has no solution (which means that the graphs are parallel), we
say that the say that the system is inconsistent.
If the graphs of the equations coincide, the system is called dependent.
If the graphs do not coincide (in other words they are parallel or they intersect), the
system is called independent.
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Section 4.2 Solving Systems of Linear Equations by Substitution
There are two algebraic methods for solving systems of linear equations. The first method
we will study is called the substitution method.
To Solve a System of Linear Equations in Two Variables Using Substitution:
1. Solve one of the equations for one of its variables.
2. Substitute the expression for the variable found in Step1 into the other equation.
3. Solve this new equation for the other variable.
4. Substitute the value obtained in Step 3 into the expression found in Step 1.
5. Write the solution as an ordered pair.
6. Check your solution by substituting the ordered pair solution into both of the
original equations.
When solving a system of linear equations algebraically, one of three possibilities will
occur:
 One ordered pair solution exists. You will obtain distinct values for x and y.
 No solution exists. When solving the system you will end up with an equation that
is never true.
 An infinite number of solutions exist. When solving the system you will end up
with an equation that is always true.
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Section 4.3 Solving Systems of Linear Equations by Addition
The second algebraic method used to solve a system of linear equations is called the
addition or elimination method.
To Solve a System of Linear Equations in Two Variables Using Elimination:
1. Write both equations in two variables in standard form ( Ax  By  C )
2. If necessary, multiply one or both equations by a nonzero number so that the
coefficients of a chosen variable in the system are opposites.
3. Add the equations.
4. Solve the resulting equation for the remaining variable.
5. Find the value of the variable eliminated in Step 3 by substituting the value you
found in Step 4 into one of the original equations.
6. Write the solution as an ordered pair.
7. Check your solution by substituting the ordered pair solution into both of the
original equations.
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Section 4.4 Solving Systems of Linear Equations in Three Variables
The standard form for a linear equation in three variables is Ax  By  Cz  D , where a,
b, c, and d are real numbers and a, b, and c are not all equal to zero.
An ordered triple (x, y, z) is a solution of a linear equation in three variables if the values
of its coordinates, when substituted for their corresponding variables, result in a true
equation.
An ordered triple is a solution of a system of linear equations in three variables if it is a
solution of every equation in the system.
To Solve a System of Linear Equations in Three Variables by Using Elimination:
1. Write all three equations in standard form.
2. Choose a pair of equations and use the equations to eliminate a variable.
3. Choose any other pair of equations and eliminate the same variable as in Step 2.
4. Form a system of two equations using the equations obtained in Steps 2 & 3 and
then solve that system using any method you wish.
5. Substitute the values obtained in Step 4 into any one of the original equations
containing the third variable, and solve for the third variable.
6. Write the solution as an ordered triple.
7. Check your solution by substituting the ordered triple solution into all three of the
original equations.
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Section 4.5 Systems of Linear Equations and Problem Solving
To model a real-world situation:
 Read and understand the problem. A picture or chart of the information may be
beneficial.
 Define variables for the unknown quantities and define other quantities in terms
of the defined variables.
 Write equations with the information given.
 Solve the system of equations algebraically.
 Check your solution by substitution.
Write an answer for the question asked, in a complete sentence.
There are many applications for systems of equations. A few of them are:
 Mixture problems
 Financial situations
 Distance, rate, and time problems
 Business and economic applications
Business and Economic Applications
The money a company spends to produce an object is called total cost, which is often
denoted by the function C x  . The money it generates upon selling the object is called
total revenue, which is often denoted by the function Rx . When one subtracts total
cost from total revenue you obtain the company’s total profit, which is often denoted by
the function Px . Thus, mathematically speaking Px  Rx  Cx .
A company is said to break-even when their total revenue is equal to total cost. In other
words, when Rx   C x  . If we solve this equation, we will obtain the x-coordinate of
the break-even point, which is the ordered pair where the graphs of the two equations
intersect each other. Once you find this x-coordinate, substitute it into either Rx or
C x  to find the y-coordinate of the break-even point and record the point as an ordered
pair with the appropriate units.
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