Transcript Systems of Linear Equations

3-1: Solving Systems by Graphing
What is a System of Linear Equations?
Definition: A system of linear equations is
simply two or more linear equations
using the same variables.
Example:
3x + 2y = 2
x + 2y = 6
Equation 1
Equation 2
How to Use Graphs to Solve Linear Systems
Consider the following system:
x – y = –1
x + 2y = 5
Using the graph to the right, we can
see that any of these ordered pairs will
make the first equation true since they
lie on the line.
y
(1 , 2)
x
We can also see that any of these
points will make the second equation
true.
However, there is ONE coordinate that
makes both true at the same time…
The point where they intersect makes both equations true at the same time.
Three Possible Outcomes
p. 154
•Two intersecting lines
•Two lines on top of each other
•Two parallel lines
EXAMPLE 4
Writing and Using a Linear System (p. 155)
EXAMPLE 4
Step 1: Write linear equations in standard form
Equation 1
y
x
1
=
+
Equation 2
y
=
2.5
x
30
EXAMPLE 4
Step 2: Graph both equations
•Two intersecting
lines = one
solution
•(20, 50) appears
to be the solution
EXAMPLE 4
Step 4: Check your solution
Point of intersection: (20,50 ).
Substitute 20 and 50 in place of x and y in both equations:
y = x + 30
y =2.5x
50 = 20 + 30
50 = 2.5(20)
ANSWER
The solution is (20, 50).
Break even point in 20 rides
Equation 1 checks.
Equation 2 checks.
Graphing to Solve a Linear System
Let's summarize! There are 4 steps to solving a linear system using a graph.
Step 1: Put both equations in slope intercept form.
Solve both equations for y, so that
each equation looks like
y = mx + b.
Step 2: Graph both equations on the
same coordinate plane.
Use the slope and y - intercept for
each equation in step 1. Be sure to
use a ruler and graph paper!
Step 3: Estimate where the graphs
intersect.
This is the solution! LABEL the
solution!
Step 4: Check to make sure your
solution makes both equations true.
Substitute the x and y values into both
equations to verify the point is a
solution to both equations.
Practice: Checking the Solution
Page 412, #11:
4x – y = 25
y
-3x - 2y = -16
We must ALWAYS verify that your
coordinates actually satisfy both
equations.
(6 , -1)
To do this, we substitute the
coordinate (6 , -1) into both
equations.
4x – y = 25
-3x - 2y = -16
4(6) – (-1) =
-3(6) - 2(-1) =
24 + 1 = 25 
-18 + 2 = -16 
Since (6 , -1) makes both equations
true, then (6 , -1) is the solution to
the system of linear equations.
x
EXAMPLE 1
Solve a system graphically
Graph the linear system and estimate the solution.
Then check the solution algebraically.
4x + y = 8
Equation 1
2x – 3y = 18
Equation 2
SOLUTION
Begin by graphing both
equations, as shown at the
right. From the graph, the lines
appear to intersect at (3, – 4).
You can check this
algebraically as follows.
EXAMPLE 1
Solve a system graphically
Equation 2
Equation 1
4x + y = 8
2x – 3y = 18
4(3) + (– 4) =? 8
2(3) – 3( – 4) =? 18
12 – 4 =? 8
8=8
6 + 12 =? 18
The solution is (3, – 4).
18 = 18
GUIDED PRACTICE
Page 153 Example 1
Graph the linear system and estimate the solution.
Then check the solution algebraically.
1. 3x + 2y = – 4
x + 3y = 1
3x + 2y = – 4
Equation 1
x + 3y = 1
Equation 2
SOLUTION
Begin by graphing both
equations, as shown at the
right. From the graph, the lines
appear to intersect at (–2, 1).
You can check this
algebraically as follows.
GUIDED PRACTICE
Equation 1
3x + 2y = –4
3(–2) + 2(1) =? –4
–6 + 2 =? –4
–4 = –4
The solution is (–2, 1).
Page 153, Example 1
Equation 2
x + 3y = 1
(–2 ) + 3( 1) =? 1
–2 + 3 =? 1
1=1
Page 142, #23
Solve the following system by graphing:
3x + 4y = -10
Start with 3x + 4y = -10
-7x - y = -10
Subtracting 3x from both sides yields
4y = –3x -10
While there are many different
ways to graph these equations, we
will be using the slope - intercept
form.
To put the equations in slope
intercept form, we must solve both
equations for y.
Dividing everything by 6 gives us…
y= -
3
4
x-
5
2
Similarly, we can add 7x to both
sides and then divide everything by
-1 in the second equation to get
y = - 7 x + 10
Now, we must graph these two equations.
Page 142, #23, cont.
Solve the following system by graphing:
y
3x + 6y = 15
–2x + 3y = –3
Using the slope intercept form of these
equations, we can graph them carefully
on graph paper.
y= -
3
4
x-
5
2
y = - 7 x + 10
Start at the y – intercept: Note that my scale is 2 on this graph. then use
the slope.
x
Page 142, #23, cont.
Can you read the solution? It looks close to (2, -4)
Check to make sure.
Graphing to Solve a Linear System
Let's do ONE more…Solve the following system of equations by graphing.
2x + 2y = 3
x – 4y = -1
Step 1: Put both equations in slope intercept form.
y = - x + 32
y = 14 x + 14
y
LABEL the solution!
(1 , 12 )
x
Step 2: Graph both equations on the
same coordinate plane.
Step 3: Estimate where the graphs
intersect. LABEL the solution!
Step 4: Check to make sure your
solution makes both equations true.
2(1)+ 2 ( 12 ) = 2 + 1 = 3
1- 4 ( 12 ) = 1- 2 = - 1