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DeSmet - Math 152
Blitzer 5E
∫ 3.1 - Systems of Linear Equations in Two Variables
1. Linear Systems:
Definition: A system of linear equations is two or more linear equations being considered at the
same time, or simultaneously. A solution to a linear system is an ordered pair that satisfies all
equations in the system. The Solution set is the set of all ordered pairs that satisfy all equations in
the system.
This section focuses on systems with 2 equations.
⎧⎪ y = x − 1
Example 1: Consider the linear system ⎨
.
y
=
3x
+
1
⎩⎪
(a) Determine if the point/ordered pair (1,0 ) is a solution to the system.
(b) Determine if the point/ordered pair ( −1,−2 ) is a solution to the system.
2. Three cases: What can happen?
Case 1: The two lines intersect at one point:
(a) A single ordered pair satisfies both equations.
(b) The solution is that ONE ordered pair, ( x, y ) , in set notation
{( x, y )} .
Case 2: The two lines are parallel with different y-intercepts.
(c)
(d)
(e)
(f)
No ordered pair satisfies both equations at the same time.
There is ____________ or we can write _____.
The system is called inconsistent.
Algebraically, we will encounter __________, for example.
Section 3.1
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DeSmet - Math 152
Blitzer 5E
Case 3: The two lines are the same line.
(g) All ordered pairs on the line(s) satisfy both equations at the same time.
(h) The solution set is all the ordered pairs on the line, {( x, y ) y = mx + b} .
(i) The system is called dependent.
(j) Algebraically, we will encounter __________, for example.
We will learn three ways of solving linear systems:
3. Method 1: Graphing
Example 2: Solve the following system by graphing. List the steps to the right.
2x = 6 − 3y
3y + x = 9
7
6
5
4
3
2
1
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
-1
-2
-3
-4
-5
-6
-7
4. Method 2: The Substitution Method: This method is best if you easily solve for one variable in
either equation.
Example 3: Solve using the substitution method
5x − 4y = 9
x − 2y = −3
Section 3.1
1. Isolate one variable in one
equation.
2. Substitute that expression
into the other equation and
solve
3. Back-substitute to find the
other variable.
4. Express your answer as an
ordered pair.
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DeSmet - Math 152
Blitzer 5E
Example 4: Solve using the substitution method
x + 3y = 9
2x − y = −10
5. Method 3: Elimination (Addition) Method: Best if you cannot easily solve for one variable.
Example 4: Solve by the elimination method.
2y − 3x = −13
3x − 17 = 4y
1. Put both equations into general
form, clearing all fractions.
2. Choose a variable to eliminate
through addition.
3. Multiply one or both equations by
the necessary constants so that the
two coefficients of the variable you
chose will eliminate as opposites.
4. Add the equations vertically.
5. Back-substitute to find the other
variable.
6. Express your answer as an ordered
pair.
Example 5: Solve by the elimination method.
3x
5
− 2y =
2
2
1
5y = − − 2x
2
Section 3.1
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Blitzer 5E
6. Example 6: Solve each system by the method of you choice. What do you find?
(a) x + 2y = 4
3x + 6y = 13
(b) y = 4x − 4
8x − 2y = 8
7. Application:
In 2010, 38.9% of U.S. males watched Jersey Shore regularly. It is predicted that in 2030, only
27.3% of U.S. males will watch Jersey Shore regularly.
In 2010, 40% of U.S. females watched Jersey Shore regularly. It is predicted that in 2030, only
24.2% of U.S. females will watch Jersey Shore regularly.
(a) Let x represent the number of years after 2010. Find linear equations (in y = mx + b ) form
that predict the percentage of male and female Jersey Shore viewers x years after 2010.
(b) In what year will the percentages be equal? What percentage of males and females are
viewers in that year?
Section 3.1
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