Download Section 7.1

MATH 1100
SECTION 7.1 Notes
Systems of Equations – Text Pages 447-453
Systems of Equations:
When solving a problem with more than one unknown, you must
have more than one equation. This set of equations is called a System
of Equations.
Note:
The number of equations should equal the number of unknowns.
Solution to a System of Equations:
A solution to a system of equations in two variables is an ordered
pair of numbers that makes both equations true.
Solving Systems of Equations:
There are three methods for solving systems of equations.
method is graphical, the other two methods are algebraic.
One
Graphical Method:
If the system of equations being solved contains equations that can
be easily graphed on a calculator, or are graphs that you can easily
sketch (like linear equations), then we can solve the system by finding
the point(s) at which all graphs in the system meet. This method gives
an estimate to the solution and is not always easy to do, so we will
concentrate on the algebraic methods of solving systems.
Substitution Method:
To use the substitution method to solve a system of equations, we
need to solve one equation for one variable and then substitute that
variable into the other equation.
Elimination Method:
To use the elimination method to solve a system of equations, use
the addition principle to cancel terms.
Example 1:
(Substitution Method)
Solve the following system of equations using the substitution
method.
2 x  y  7

3x  y  13
solve one equation for a variable:
substitute into the other equation and solve for the other variable:
substitute solution into the initial equation and solve for the
remaining solution:
Example 2:
(Substitution Method)
Solve the following system of equations using the substitution
method.
 x2  y  9

x  y  3  0
solve one equation for a variable:
substitute into the other equation and solve for the other variable:
substitute solution into the initial equation and solve for the
remaining solution:
Example 3:
(Elimination Method)
Solve the following system of equations using the elimination
method.
 4 x  3 y  10

9 x  4 y  1
multiply entire equations by appropriate values and add the two
equations together:
solve for the remaining variable:
plug solution into one of the original equations and solve for other
variable:
check and state:
Example 4:
(Elimination Method)
Solve the following system of equations using the elimination
method.
3 x 2  4 y  17
 2
2 x  5 y  2
multiply entire equations by appropriate values and add the two
equations together:
solve for the remaining variable:
plug solution into one of the original equations and solve for other
variable:
check and state:
Example 5:
Solve the following system of equations with the elimination
method and with the substitution method.
 x  y  180

x  2y  5

Substitution Method:
Elimination Method:
Example 6:
Find the lengths of the sides of a right triangle if the length of the
hypotenuse is 25 ft and the area is 84 ft2.