Mod 7 notes
Systems of Equations
-two or more equations with 2 or more variables
-usually the number of equations equals the number of variables
Solution to a System of Equations
-value(s) of variables must satisfy all equations
Recall linear equations:
2 x  2  14
2x 12
x  6 ( one solution)
Whereas x  y  6 has 2 variables having many solutions {(1,5), (2,4), ( 3,3)} are just
a few.
Example:
1. Which of the following is a solution to the system y = 3x + 5 and 6x  5y = 43?
a. (1, 8)
b. (5, 3)
c. (2, 11)
d. (2, 11)
Answer D (2, 11)
For a solution to be valid, it must satisfy both equations simultaneously.
i.e. 1,8 works in the first equation but not the second.
Systems of Linear Equations: Graphing
-To graph a system of equations, graph each line on the same coordinate plane:
A) If the lines are intersecting, find the point of intersection; the ordered pair is the
solution.
B) If the lines are parallel, there is no solution.
C) If the lines coincide, there are infinitely many solutions.
Example:
Systems of Linear Equations: Using your Graphing Calculator
Example: By using a graphing calculator, find the point of intersection for
y = 2x + 2
y = -x - 4
Before you begin, you need to clear the memory of your calculator. Use the following
buttons in your TI84.
You are now ready to begin.
Solving a System of Linear Equations: Substitution
-It is not always convenient to solve systems of equations by graphing. You may not
have access to graphing paper or a graphing calculator.
Solving Systems Using Substitution
1. Solve either one of the equations for one variable in terms of the other.
2. Substitute the expression you just found into the other equation.
3. Solve for the variable.
4. Substitute the value into the equation used in step one to find the value of the
other variable.
5. Check your answer by substituting the values into the equation not used in
step one.
Example: Find the solution for the system:
This shows that we have the correct solution. Please note that if this question were
graphed, the intersection of these two graphs would be located at (-1,-2) as shown
in diagram below:
Example: Find the solution for the system:
This is a false statement, which means that there is no solution.
Example: Find the solution for the system:
This is a true statement, which means that there are infinitely many solutions.
Note: If you were to change both equations into slope-intercept form, you would get
y = -3x + 8. Because they are the same line, they coincide and have infinitely many
solutions.
Solving Systems Using Elimination
-Use the following steps to solve a system of equations by elimination.
1) If necessary, rewrite both equations in standard form (Ax + By = C).
2) Choose a variable to eliminate. If needed, multiply one or both equations by a
constant so that when you combine the equations, the variable you chose will be
eliminated.
3) Solve for the remaining variable.
4) Substitute this value into one of the equations to find the other variable.
5) Check your solution by substituting into both equations.
Example: Find the solution for the system:
Example: Find the solution for the system:
Both equations are in Ax + By = C form.
Eliminate the x terms by multiplying the second equation by 2.
Because both equations are now the same, everything is eliminated. The lines are
coincidental and there is an infinite number of solutions.
Modeling situations using A system of Linear Equations
-Expressions and equations are very similar.
-An equation is a mathematical sentence that says that two things are equal. An
equation always contains an equal sign. It can contain numbers, variables,
operations, or some combination of numbers, variables, and operations. For
examples, the following are equations:
5n + 1 = 21
n - 1 = 5
-An expression is a mathematical phrase that can be a number, variable, or
combination of numbers, variables, and operations. For example, the following are
algebraic expressions:
14n – 1
5n
-21
In all these examples, “n” represents the unknown. The letter “n” or any letter used
to represent the unknown is called a variable.
-To transform words into mathematical symbols, you need to understand the
following translations and examples.
Addition
The + sign can be represented with the following word phrases. (Notice that several
variables are used; any variable can be used to represent a number.)
1) Added to
Example: A number added to six translates into 6 + n.
In addition, order does not
matter, so this could also be expressed as n + 6.
2) Increased by
Example: Twenty-one is increased by a number translates into 21 + r.
3) Sum
Example: The sum of a number and twelve translates into w + 12.
4) More than
Example: One more than a number translates into c + 1.
5) Plus
Example: Five plus a number translates into 5 + v.
Multiplication
The x sign can represent multiplication in the following word phrases.
1) Multiplied by
Example: A number multiplied by three translates into n x 3.
In multiplication,
order does not matter, so this could also be 3 x n or, in a more simplified form, 3n.
2) Product
Example: The product of four and a number translates into 4 x n or 4n.
3) Times
Example: Nine times a number translates into 9 x e or 9e.
4)Twice as much, three times as much, four times as much... etc.
Example: Twice as much as a number translates into 2 x g or 2g.
To review, in
multiplication and addition, order does not matter. In other words, n + 5 is the
same as 5 + n and 4 x d is the same as d x 4, but we would usually write this as 4d.
This information is defined in two properties:
Commutative property of addition a + b = b + a
Commutative property of multiplication a x b = b x a
In subtraction and division, order does matter and, therefore, you must be very
careful when converting English phrases into mathematical expressions to ensure
that you have the order correct.
Subtraction
The subtraction (–) sign can be represented with the following word phrases.
1) Decreased by
Example: A number decreased by three translates into n – 3. (This is not the same
as 3 – n)
2) Minus
Example: Four minus a number translates into 4 – n.
3) Less than or fewer than
Example: Nine less than a number translates into r - 9.
4) Difference between
Example: The difference between 2 and a number translates into 2 – f.
Division
The division (÷) sign can be represented with the following word phrases.
1) Out of
Example: Six out of a number translates into
2) Divided by
Example: Nine divided by a number translates into 9 ÷ x. (Again, this is not the
same as x ÷ 9.)
Equals
All the above examples represent expressions. They could easily become equations
with one simple ingredient, the equal sign (=). The equal sign can be represented
with English words such as the following:
1) Is
Example: Nine divided by a number is four translates into 9 ÷ x = 4.
2) Gives
Example: Five minus a number gives one translates into 5 – n = 1.
3) Yields
Example: The product of seven and a number yields forty-two translates into 7 x n =
42.
Word Problems
Word problems can often be solved using mathematical equations. This lesson
focuses on word problems that can be translated into a system on linear equations.
The word problems you will deal with will have two variables and two unknowns.
Example: The sum of two numbers is 120. The difference between the
numbers is 40. Find the numbers.
Step 1: Construct let statements. Read the problem and determine the unknowns
the question is asking you to find. These become your "let" statements.
The example above asks you to “find the numbers”. Therefore, the "let"statements
would look like this:
Let x = the larger number
Let y = the smaller number
Step 2: Write two equations with two unknowns that reflect the information in the
question. Use the variables described in your "let" statements in Step 1.
"The sum of two numbers is 120" translates into
x + y = 120
"The difference between the numbers is 40" becomes
x – y = 40
Step 3: Solve the linear system using graphing, substitution, or elimination. In this
example you are given one positive y and one negative y, so use the elimination
method. If the question does not specify, always use the method that you think will
be the easiest based on how the equations are set up.
Step 4: Use one of the equations from step 2 to find the other unknown. Because
you know that x = 80 and you know that x + y = 120, you get the following:
Step 5: Answer the word problem with a word sentence.
The two numbers are 40 and 80.
Step 6: Check your answers to ensure that they make sense.
80 + 40 = 120 and 80 – 40 = 40
Therefore, the answers are correct.
Example 2
The Sum of two numbers is 15 and their difference is 3. What are the numbers?