Avon High School
Section: 8.1
ACE COLLEGE ALGEBRA II - NOTES
Systems of Equations in Two Variables
Mr. Record: Room ALC-129
Semester 2 - Day 1
Systems of Linear Equations and Their Solutions
Often in mathematics, it becomes interesting to determine the intersection points of
two (or more) graphs. In this section we will focus on the graphs of two lines.
Note the intersection point of the two linear equations graphed to the right.
But sometimes, we may be interested in determining a point of intersection without
using the time to sketch a graph.
Example 1
Determining Whether Ordered Pairs Are Solutions of a System
Consider the system
2 x  3 y  4
2x  y  4
. Determine if each ordered pair is a solution of the
system.
a. 1, 2 
b.
 7, 6 
Solving Linear Systems: The Substitution Method
Solving Linear Systems by Substitution
1. Solve either of the equations for one variable in terms of the other. (If one of the equations
is already in this form, you can skip this step.)
2. Substitute the expression found in Step 1into the other equation. This will result in an
equation in one variable.
3. Solve the equation containing one variable.
4. Back-substitute the value found in Step 3 into one of the original equations.
Simplify and find the value of the remaining variable.
5. Check the proposed solution in both of the system’s given equations.
Example 2
Solving a System by Substitution
Solve by the substitution method.
3x  2 y  4
2x  y  1
Let
be positive real numbers with
, and let p be any real number.
The logarithm of a number with an exponent is the product of the exponent and the logarithm
of that number.
Solving Linear Systems: The Addition Method (Eliminating a Variable)
Solving Linear Systems by Addition
1. If necessary, rewrite both equations in the form: Ax  By  C .
2. If necessary, multiply either equation or both equations by appropriate nonzero numbers so
that the sum of the x-coefficients or the sum of the y-coefficients is 0.
3. Add the equations in Step 2. The sum is an equation of one variable.
4. Solve the equation obtained in Step 3.
5. Back-substitute the value found in Step 4 into one of the original equations.
Simplify and find the value of the remaining variable.
6. Check the proposed solution in both of the system’s given equations.
Solving a System by the Addition Method
Example 3
Solve by the addition method.
4x  5 y  3
2x  3y  7
Let
be positive real numbers with
, and let p be any real number.
Example
4
“Messy”
System
byproduct
the Addition
Method
The
logarithm
ofSolving
a numberawith
an exponent
is the
of the exponent
and the logarithm
Solve
by
the
addition
method.
of that number.
2x  9  3y
4 y  8  3x
Linear Systems Having No Solution or Infinitely Many Solutions
The Number of Solutions to a System of Two Linear Equations
The number of solutions to a system of two linear equations in two variables is given by one of
the following:
What This Means Graphically
Number of Solutions
What This Looks Like
Exactly one ordered pair
No Solution
Infinitely many solutions
Example 5
Example 6
Solving a Linear System - Unusual Result
Solve the system.
5x  2 y  4
10 x  4 y  7
Solving a Linear System - Unusual Result
Solve the system.
x  4y 8
5 x  20 y  40
Functions of Business: Break-Even Analysis
Revenue and Cost Function
A company produces and sells x units of a product.
Revenue Function R( x)  (price per unit) x
Cost Function C ( x)  fixed cost + (cost per unit produced) x
Example 7
The point of intersection of the graphs of
revenue and cost functions is called the
break-even point. The x-coordinate of
that point reveals the number of item to
be produced and sold where the ycoordinate is equal to the amount of
money going in or out.
Finding a Break-Even Point
A company that manufactures running shoes has a fixed cost of $300,000. Additionally, it
costs $30 to produce each pair of shoes. They are sold at $80 per pair.
a. Write the cost function, C, of producing x pairs of running shoes.
Write the
R,, from
thep sale
of x pairs of running shoes.
Let
beb.positive
realrevenue
numbersfunction,
with
and let
be any
real number.
c. Determine the break-even point. Describe what it means.
The logarithm of a number with an exponent is the product of the
Profit Function
exponent and the logarithm
The profit, P(x), generated after
of that number.
d. Write the profit function for shoe company
producing and selling x units of a
above.
product is given by the profit function.
P( x)  R( x)  C ( x)
where R ( x ) and C ( x) are the revenue
and cost functions respectively.