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Lesson 24 MA 152, Section 6.1
A linear equation with two variables has infinite solutions (many ordered pairs, example
2 x  3 y  6 ) However, if a second equation is also considered, and we want to know
what ordered pairs they may have in common, how many solutions may there be? When
two linear equations are considered together, it is known as a system of equations. The
solution of the system if any ordered pair(s) they have in common.
Consider graphing two lines. How many points may they have in common?
As demonstrated in the graphs above, there are 3 situations.
1. The graphs may intersect at one point. If the lines of the equations intersect, the
solution is one ordered pair.
2. The graphs may intersect at an infinite number of points. If the lines of the
equations form the same line, the solution is an infinite number of ordered pairs.
3. The graphs may not intersect. If the lines of the equations do not intersect, there
is no solution.
There are three methods that may be used to find the solution of a system of linear
equations. You already know one method. Two more are in this lesson.
1. The Graphing Method-Graph each line and identify any common point, or
determine if the lines are the same or if the lines are parallel.
2. The Substitution Method
3. The Addition Method
For the first example we will use the graphing method. The graphing method is not
really the best method. Graphing is not very accurate. And, if the solution has
coordinates that are fractions or decimals, the coordinates of the point of intersection (if
there is one) would be difficult to determine.
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 3x  2 y  2
Ex 1: Find the solution to the system 
by graphing.
2 x  3 y  16
Substitution Method:
1. Solve on equation for a variable, if necessary.
2. Substitute the expression obtained for that variable for that variable in the second
equation.
3. Solve the equation that results for that variable.
4. Substitute the solution found in step 3 into any of the equations and solve for the
remaining variable.
5. Write the solution as an ordered pair.
Ex 2: Solve each system of linear equations using the substitution method.
2 x  3 y  0
a) 
 y  3x  11
2
b)
 4x  5 y  4

8 x  15 y  3
Examine these examples.
 x  3 y  4
1
4
A 
y  x

3
3
4
 1
x  3  x    4
3
 3
4
 1
2 x  6  x    8
3
 3
2x  2x  8  8
88
B
 y  x  3

 x y 7
x  ( x  3)  7
x x3 7
37
You will notice that both variables 'dropped out' in both example A and example B.
In example A, a true statement was the result (8 = 8). This indicates that there are several
solutions, because the statement is true. Graphically, these are the same line. There are
an infinite number of solutions, every point on the line. The general solution may be
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4

written as an ordered pair this way:  x, x   .
3
3

A system of equations that has a solution is called consistent. Example A is a consistent
system. However, if the equations lead to the same graph, the solution is also called
dependent. Example A is also dependent. (One of the equations in a dependent system
can be found by multiplying both sides of the other equation by a nonzero constant.)
In example B, the result was a false statement (3 = 7). Because the result was false, there
is no ordered pair that is common to both equations. The system has no solution. This
system is not consistent, but is inconsistent (no solution).
A system where the equations are not equal is also called independent. Both systems in
example 2 are consistent and independent.
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3x  y  12
Ex 3: Solve the system 
. Describe the solution and define the system using
 y  3x  12
the words consistent, inconsistent, dependent, or independent.
Addition Method: As with the substitution method, the goal of the addition method is to
eliminate one of the variables.
1. Write both equations in general form so that terms with the same variable are
aligned vertically.
2. Multiply all the terms of one or both of the equations by constants chosen to make
the coefficients of one variable differ only in sign.
3. Add the equations. Solve the equation that results.
4. Substitute the value you found in step 3 into any of the equations and solve for the
remaining variable.
5. Write the solution as an ordered pair.
Ex 4: Solve each system. Write a description for each system.
2 x  3 y  8
a) 
 5x  y  3
b)
3x  3 y  y  9

 5( x  y )  15
4
c)
 3x  4(2  y )

3x  6  4 y  0
Many application problems can be solved by using a system of equations. Usually the
addition method or the substitution method is used to solve the problem.
1.
2.
3.
4.
Define two variables for the problem.
Write two equations representing statements from the problem.
Solve using either the substitution method or the addition method.
Answer the question posed in the problem.
Ex 1: A rectangular soccer field has a perimeter of 360 yards. Its length is 20 yards more than
its width. What are the dimensions of the field?
w = width
L = length
Ex 2:
During the 96-97 NBA season, the Boston Celtics played 82 games. They lost 52 more
games than they won. What was their record that year?
w=
L=
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Ex 3:
In his motorboat, Nigel travels upstream at top speed to his favorite spot, a distance of 36
miles, in two hours. Returning downstream, still at top speed, the trip only takes 3/2
hours. Find the top speed of the boat (as if it was in still water) and the speed of the
current.
Rate
Time
Distance
upstream
downstream
Ex 4: A manufacturer of CDs have determined that month fixed costs to make the CDs
are $337,500. It costs $3.50 to make a CD. Each CD that is made can be sold
for $9.75. Find an equation E to represent the expenses of the manufacturer and
an equation R to represent to revenue where x represents the number of CDs made
and sold during a month. At what production level will expenses equal revenues
(break-even point)?
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Ex 5: Marjorie has been hired as an advertisement salesperson by a local media station.
She is offered two options for her salary. In option A, she is paid $350 weekly
plus 20% of her sales. In option B, she is paid $270 weekly plus 22% of her
sales. Write equations SA and SB to represent her salaries from each option, if x
equals her sales for the week. How much must Marjorie's sales total so these
plans would equal the same salary?
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