Download Section 3.1 and 3.2: Solving Linear Systems by Graphing

• If given a solution for an equation, you can
plug in the solution’s number to see if the
numbers match your equation.
Ex: Plug in the solution (2,1) to the given
equations:
• 2x-3y=1
2(2) – 3(1)=1 1=1
• x+y=3
2+1=3
3=3
Checking Solutions of a Linear System
Try on your own:
Ex: (-2 , -8)
2x – y=52
9x – y=-10
2(-2)–(-8)=52
-4+8=52 ; -4≠52
(-2,-8) Isn’t a solution to the equation
9(-2)—(-8)=-10
-18+8=-10 ; -10 =-10 (-2,-8) Is a solution to the equation
Ex:
x+2y=9
-x+6y=-1
1. Put the two equations into slope-intercept form (y=mx+b)
2y=-x+9
6y=x-1
y=-x/2 + 9/2
y=x/6 -1/6
2. Graph the new equations.
3. When the two equations are graphed,
a point of intersection is made. For
this example, (7,1) is the point of
intersection.
4. Plug in the point of intersection into
the original equations to see if the
solution works.
• You perform the same steps as before,
however the graphs will look different.
• Equations with infinitely many solutions
will share the same line on the graph.
• Equations with no solutions will form
parallel lines on a graph.
Ex: 3x-2y=6
6x-4y=12
Y=3x/2-3
Y= 3x/2-3
Since the equations share a line, each
point on the line is a solution. Therefore
there is an infinite amount of solutions.
Ex: 3x – 2y=6
3x-2y=2
Y= 3x/2 – 3
Y= 3x/2 – 1
Since the lines are II there is no point
of intersection, therefore there is no
solution to the equations.
There are 2 methods to solving a linear system
algebraically:
Substitution Method
1. Solve for 1 of the variables in 1 of the equations
2. Substitute the expression from step 1 into the
second equation to solve for the other variable
3. Substitute the value from step 2 into the revised
equation from step 1 and solve.
Ex: 3x+4y=-4 ; x+2y=2
1. Solve for x in one the equations:
x= -2y+2
2. Substitute the expression into the other equation
and solve for y:
3(-2y+2) + 4y=-4
-6y+6+4y=-4
-2y+6=-4 -2y=-10 y=5
3. Substitute y into your revised equation from step 1.
x= -2(5)+2 x=-10+2 x=-8 y=5 (-8,5)
*Check to see if your solution is correct by plugging in the
numbers to your original equations.
The second method is:
Linear Combination:
Ex: 2x-4y=13
4x-5y=8
1. Multiply 2x-4y=13 by -2. By doing this, both coefficents in the
equations will be 4, only their signs will differ:
-4x+8y=-26
4x-5y=8
2. Add these two equations together:
The coefficents -4 and 4
cancel out when added
together.
-4x+8y=-26 3y=-18 y=-6
4x -5y=8
3. Substitute y into the original equation to solve for x:
2x-4(-6)=13
2x+24=13
2x= -11
x= -11/2
(-11/2, -6) is your solution
*Check to see if your solution is correct by plugging the numbers
into your original equations
• Use the same steps in the linear combination method.
After multiplying the first equation by 2, add the two equations
together:
This equation has infinitely many
14x+ -14x + 4y-4y =0
0=0 solutions
After multiplying the first equation by 3, add the two equations
together:
This equation has no
18x + -18x +-3y + 3y=-2 0≠-2 solutions.
Try It On Your Own
Graph: 3x+4y = -10
-7x+ -y = -10
Y=-3x/4 – 5/2
Y= -7x+ 10
(2,-4) Is the
point of
intersection.
3(2) + 4 (-4) = -10 6 + -16= -10 -10=-10
-7 (2) – (-4) = -10 -14+ 4 = -10 -10=-10
How many solutions are there to this equation?
One
Try It On Your Own
Graph: 7x + 2y= 16
-21x + 6y= 24
Y= -7x/2 + 8
Y= -7x/2 - 4
How many
solutions are there
to the equations?
There are no solutions to the
equations
Try It On Your Own
Solve Using the Substitution Method:
Ex: x+ 2y =2
7x – 3y = -20
X= -2y + 2
7(-2y + 2) -3y=-20
-14y + 14 – 3y =-20
-17y=-34
Y=-34/-17
Y=2
X=-2(2) + 2
X= -2
Try It On Your Own
Solve by Using the Linear Combination Method
Ex: 3x – y = 4
9x – 3y = 12 9x – 3y = 12
-9x + 3y = -12
-9x + 3y = -12 -9x + 3y = -12
0=0
How many
solutions are there
for these
equations?
Infinitely many
solutions
Try It On Your Own
Solve by Using the Linear Combination Method
Ex: 3x – y =7
2x + 3y =1
9x – 3y = 21
2x + 3y =1
9x – 3y = 21
2x + 3y =1
11x=22
x=2
3 (2) – y =7
6 – y =7
-y = 1
Y= -1
How many solutions are
there to the equations?
One