Download Systems of Equations in Two Unknowns

Systems of Two Equations in Two Unknowns
• In order to explain what we mean by a system of equations, we
consider the following:
Problem. A pile of 9 coins consists of nickels and quarters. If the
total value of the coins is $1.25, how many of each type of coin are
there?
Solution. Let x = the number of nickels and let y = the number of
quarters. The requirements of the problem can be expressed as:
x  y  9
5 x  25 y  125.
This is an example of a system of two equations in two unknowns,
and we want a pair of numbers (in this case x = 5, y = 4) which
satisfy both equations. Such a pair of numbers is called a solution of
the system.
Solution by Substitution
• Suppose we have a system of two equations in two unknowns. If
we can use one of the equations to express one variable in terms
of the other variable, then we can substitute this expression into
the second equation, which results in a single equation
containing a single unknown.
• For the system from the previous slide, we solve by substitution:
y  9 x
5 x  25(9  x)  125
 20 x  225  125
 20 x  100
x  5, y  9  x  4.
Another Example for Solution by Substitution
• Solve the system of equations:
x 2  y 2  25
x  y  1.
• Solution.
y  1  x
x 2  (1  x) 2  25
x 2  x  12  0
( x  4)( x  3)  0
x  4 or x  3
y3
y  4
• Note: In this example, the system has two solutions.
Solution by Graphing
• If we graph the two equations of a system, then the coordinates
of the points of intersection of the two graphs must be the
solutions of the system.
• Example from previous slide:
x 2  y 2  25
(4,3)
(3,4)
x  y  1
Consistent and Inconsistent Systems
• A consistent system has one or more solutions. We have seen
examples of consistent systems in previous slides.
• An inconsistent system has no solutions. Geometrically, this
happens when the two graphs have no points of intersection.
• Example. The system given below is inconsistent. Do you see
why?
x  y 1
2 x  2 y  3.
Solution by Elimination
• The method of elimination seeks to combine the equations of a
system in such a way to eliminate one of the unknowns.
• Example. Solve the system:
x 2  y 2  25
x  y2  5
x2  x
 20 Subtract 2nd equation from 1st.
x 2  x  20  0
( x  5)( x  4)  0
x  5 or x  4
Now, substitute the values for x into one of the original equations and
solve for y. This results in three solutions: (5, 0), (4, 3), (4,  3).
Systems of Linear Equations
• A system consisting only of equations that are of first degree
in x and y is called a system of linear equations or simply a
linear system. When we graph a system of two linear
equations, we are graphing two straight lines. There are 3
possibilities:
1. The two lines intersect at a point. The system is consistent
and has a unique solution—the point of intersection.
2. The two equations are different forms of the same line. The
system is consistent and has an infinite number of solutions—
all points on the line.
3. The two lines are parallel. Since the lines do not intersect,
the system is inconsistent and has no solution.
Solving an Inconsistent System by Elimination
• Problem.
• Solution.
x  y 1
2 x  2 y  3.
 2 x  2 y  2 Multiply first equation by  2.
2 x  2 y  3.
0  1. Add the equations.
Since assuming that a solution exists
leads to a contradiction (0 = 1), we
conclude that no solution exists.
Summary of Systems of Eqns in Two Unknowns; We discussed
• Solution of a system as a pair of numbers
• Solution by substitution
• Solution by graphing
• Consistent and inconsistent systems
• Solution by elimination
• Systems of two linear equations and their solutions