Good morning everybody.
we will take you on a fun
learning today.
New generation S.W.K
By:
Mrs. Panchaporn Kantayasakun
Mrs. Khrongsri Nampoon
Mrs. Jarin Promsri
System of Linear Equations
How to: solve by graphing, substitution, linear
combinations, and special types of linear systems
What is a Linear System,
Anyways?
• A linear system includes two, or more,
equations, and each includes two or more
variables.
• When two equations are used to model a
problem, it is called a linear system.
Before You Begin…Important
Terms to know
• Linear system: two equations that form one
equation
• Solution: the answer to a system of linear
equation; must satisfy both equations
***: a solution is written as an ordered pair:
(x,y)
• Leading Coefficient: any given number that is
before any given variable (for example, the
leading coefficient in 3x is 3.)
• Isolate: to get alone
Solving Linear Systems by
Substitution
• Basic steps:
• 1. Solve one equation for one of its
variables
• 2. Substitute that expression into the other
equation and solve for the other variable
• 3. Substitute that value into first equation;
solve
• 4. Check the solution
Example: The Substitution
Method
• Here’s the problem:
Equation one -x+y=1
Equation two 2x+y=-2
First, solve equation one for y
y = x+1
Next, substitute the above expression in for “y” in
equation two, and solve for x
Here’s how:
Equation two
2x+y
=
-2
2x+ (x+1) =
-2
3x+1
=
-2
3x
=
-3
x =
-1
Congratulations! You now know x has a value of –1…but
you still need to find “y”.
To do so…
First, write down equation one
y =
x+1
y =
(-1)+1
y =
0
So, now what?
You’re done; simply write out the solution as (-1,0)
***Did you remember?
To write a solution, once you’ve found x and y, you
must put x first and then y: (x,y)
Solving Linear Systems by Linear
Combinations
Solving Systems by means of
Linear Combinations
• Basic steps:
1. Arrange the equations with like terms in columns
2. After looking at the coefficients of x and y, you
need to multiply one or both equations by a number that
will give you new coefficients for x or y that are opposites.
3. Add the equations and solve for the unknown
variable
4. Substitute the value gotten in step 3 into either of
the original equations; solve for other variable
5. Check the solution in both original equations
Example: Solving Systems by
Linear Combinations
• Solve this linear equation:
Equation One: 3x+5y = 6
Equation Two: -4x+2y =5
Solve the linear system
Equation 1: 3x+5y=6
Equation 2: -4x+2y=5
3x+5y
= 6 --------
-4x+2y
= 5 --------
4;
4(3x+5y) = 46
12x+20y = 24 --------
3 ;
3(-4x+2y) = 35
-12x +6y = 15 --------
+ ;
12x+ 20y -12x + 6y = 24 + 15
26y = 39
Equation 2: -4x+2y=5
Substitute the value you just found for
-4x+2( ) = 5
-4x+3 = 5
-4x = 2 1
x = 2
The solution to the example system is
13
( , )
22
A Final way to Solve Systems:
Graph and Check
Types of Solutions of Systems of Equations
• One solution – the lines cross at one point
• No solution – the lines do not cross
• Infinitely many solutions – the lines coincide
An Example of the Quick graph
on and Check Method
• Here’s the problem:
Equation one -x+y=1
Equation two 2x+y=-2
Step 1 Download Application Quick graph from
programe App Store.
Step 2 open App Quick graph on Iphone or
samsung etc.
Step 3 Click the plus sign.
Step 4 Type the equation in the form
y=1+x and click Done .
Step 5 will have a graph
Step 6 Click the plus sign. And Type
the equation in the form y=-2-2x
and click Done
Step 7 will have a graph for
equation y=-2-2x
Answer to the
equation is the graph
intersect (-1,0)
Fun, Fun: Exercises
• 1. Solve the following Linear System
Equation one: 3x-4y=10
Equation two: 5x+7y=3
• 2. Solve the following Linear system
Equation one: x-6y=-19
Equation two: 3x-2y=-9
• 3. Solve the following Linear system
Equation one: x+3y=7
Equation two: 4x-7y=-10
• 4. Use linear combinations to solve this
system
Equation one: x+2y=5
Equation two: 3x-2y=7
• 5. Use linear combinations to solve this
system
Equation one: 3x-5y=-4
Equation two: -9x+7y=8
Answers to the Exercises
•
•
•
•
•
1. (2,-1)
2. (-1,3)
3. (1,2)
4. (3,1)
5. (-0.5, 0.5)
Check out the answers from of
the Quick graph
1.
3x-4y=10
5x+7y=3
Check out the answers from of
the Quick graph
2.
x-6y=-19
3x-2y=-9
Check out the answers from of
the Quick graph
3.
x+3y=7
4x-7y=-10
Check out the answers from of
the Quick graph
4.
x+2y=5
3x-2y=7
Check out the answers from of
the Quick graph
5.
3x-5y=-4
-9x+7y=8