Math
1314 College
Algebra LO-9 Chapter 6.1
Skills
to Master
·
Solve systems of
equations in two variables by graphing
·
Solve systems of
equations in two variables by substitution
·
Solve systems of
equations in two variables by the addition method
·
Determine when a
system in two variables has infinitely many solutions and relate such a system
to a graph of the equations
·
Determine when a
system in two variables is inconsistent and relate such a system to a graph of
the equations
Systems
of Equations
SA DR Instructor Student
We are interested in
the distance from San Antonio over time.
That is, the distance each individual travels over time is a function of
elapsed time. So we can write an
equation for each relating distance to time, where the speed is the rate of
change (i.e., the slope) and the starting point as shown above is the
y-intercept. If we use d to stand for distance in miles and t to stand for time in hours, then:
Student:
A graph of these two
linear equations is shown at right:
A system of equations is a set of equations (or functions) that may
or may not have some number of points in common. Solving a system means finding the common
points (if any).
Methods
·
Solve by
graphing
·
Solve by
substitution
·
Solve by the
addition method
Solving
by Graphing
The solution is the point of intersection of the two lines.
Remember that there are
other ways to graph a line if they are not in slope-intercept form.
Solving
by Substitution
Remember that, if two
values are equal, one can be substituted in place of the other. For example, if
The same is true even
if what is to be substituted is an expression.
For example, if
Example
Remember that solving
the system means finding a point. So you must find an x-coordinate value AND a
y-coordinate value.
Solving
by the Addition Method
Solving by the addition
method involves combining like terms from a pair of equations. The goal is to adjust one or both equations
such that one of the variables will cancel to zero.
Example
The previous examples
were easy because the terms were already opposites of each other. What if they are not opposites to begin with?
Example
Example
Suppose the equations
are such that changing only one will not cancel terms?
Systems with infinitely
many solutions
Solve the system
Why does this
happen? Look at the graphs of the two
equations:
This system has an
infinite number of solutions because the two equations are actually the same,
one on top of the other.
Inconsistent systems
Solve the system
What does the graph of
this system look like?
The system is inconsistent which means that there are
no solutions. Graphically, this means
that the lines are parallel so there is no point of intersection.