systems equations two varibles
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Systems of Linear Equations
Using a Graph to Solve
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How to Use Graphs to Solve Linear Systems
x
yConsider the following system:
x – y = –1
x + 2y = 5Using 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.
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…
(1 , 2)
The point where they intersect makes both equations true at the same time.
•If the lines cross once, there will be one solution.
•If the lines are parallel, there will be no solutions.
•If the lines are the same, there will be an infinite number of solutions.
Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, determine the solution.
1 3 3
3 9 9
.
x y
x y
23
54
5 3
. y x
y x
3 3
2 6
. x y
x y
x
y The two equations in slope-intercept form are:
y x
y x o r y x
1
31
3
9
9
9
1
31
Plot points for each line.Draw in the lines.
These two equations represent the same line.
Therefore, this system of equations has infinitely many solutions .
The two equations in slope-intercept form are:
x
y
y x
y x
3
54
3
5
Plot points for each line.
Draw in the lines.
This system of equations represents two parallel lines.
This system of equations has no solution because these two lines have no points in common.
x
yThe two equations in slope-intercept form are:y x
y x
3
2 6
Plot points for each line.
Draw in the lines.
This system of equations represents two intersecting lines.
The solution to this system of equations is a single point (3,0) .
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.
Step 2: Graph both equations on the same coordinate plane.
Step 3: Estimate where the graphs intersect.
Step 4: Check to make sure your solution makes both equations true.
Solve both equations for y, so that each equation looks like
y = mx + b.
Use the slope and y - intercept for each equation in step 1. Be sure to use a ruler and graph paper!
This is the solution! LABEL the solution!
Substitute the x and y values into both equations to verify the point is a solution to both equations.
Graphing is not the only way to solve a system of equations. It is not really the best way because it has to be graphed perfectly and some answers are not integers.
SOOOOWe need to learn another way!!!!
Solve: by ELIMINATION
x + y = 12-x + 3y = -8
We need to eliminate (get rid of) a variable.
The x’s will be the easiest. So, we will add the two equations.
4y = 4 Divide by 4
y = 1
THEN----
Like variables must be lined under each other.
X +Y = 12
(11,1)
Substitute your answer into either original equation and solve for the second variable.
Answer
Now check our answers in both equations------
x + 1 = 12 -1 -1 x = 11
X + Y =12
11 + 1 = 12
12 = 12
-x + 3y = -8
-11 + 3(1) = -8
-11 + 3 = -8
-8 = -8
Solve: by ELIMINATION
5x - 4y = -21-2x + 4y = 18
We need to eliminate (get rid of) a variable.
The y’s be will the easiest.So, we will add the two equations.
3x = -3 Divide by 3
x = -1
THEN----
Like variables must be lined under each other.
5X - 4Y = -21
(-1, 4)
Substitute your answer into either original equation and solve for the second variable.
Answer
Now check our answers in both equations------
5(-1) – 4y = -21 -5 – 4y = -21
5 5-4y = -16
y = 4
5x - 4y = -215(-1) – 4(4) = -21
-5 - 16 = -21-21 = -21
-2x + 4y = 18
-2(-1) + 4(4) = 18
2 + 16 = 18
18 = 18
Solve: by ELIMINATION
2x + 7y = 315x - 7y = - 45
We need to eliminate (get rid of) a variable.
The y’s will be the easiest. So, we will add the two equations.
7x = -14 Divide by 7
x = -2
THEN----
Like variables must be lined under each other.
2X + 7Y = 31
(-2, 5)
Substitute your answer into either original equation and solve for the second variable.
Answer
Now check our answers in both equations------
2(-2) + 7y = 31 -4 + 7y = 31
4 47y = 35
y = 5
2x + 7y = 312(-2) + 7(5) = 31
-4 + 35 = 3131 = 31
5x – 7y = - 45
5(-2) - 7(5) = - 45
-10 - 35 = - 45
- 45 =- 45
Solve: by ELIMINATION
x + y = 30 x + 7y = 6
We need to eliminate (get rid of) a variable.
To simply add this time will not eliminate a variable. If one of the x’s was negative, it would be eliminated when we add. So we will multiply one equation by a – 1.
Like variables must be lined under each other.
X + Y = 30
X + 7Y = 6( ) -1
X + Y = 30
-X – 7Y = - 6
Now add the two equations and solve.
-6Y = 24
- 6 - 6
Y = - 4
THEN----
X + Y = 30
(34, - 4)
Substitute your answer into either original equation and solve for the second variable.
Answer
Now check our answers in both equations------
X + - 4 = 30 4 4
X = 34
x + y = 3034 + - 4 = 30
30 = 30
x + 7y = 6
34 + 7(- 4) = 6
34 - 28 = 6
6 = 6
Solve: by ELIMINATION
x + y = 4 2x + 3y = 9
We need to eliminate (get rid of) a variable.
To simply add this time will not eliminate a variable. If there was a –2x in the 1st equation, the x’s would be eliminated when we add. So we will multiply the 1st equation by a – 2.
Like variables must be lined under each other.
X + Y = 4
2X + 3Y = 9
-2X - 2 Y = - 8
2X + 3Y = 9
Now add the two equations and solve.
Y = 1
THEN----
( ) -2
(3,1)
Substitute your answer into either original equation and solve for the second variable.
Answer
Now check our answers in both equations------
X + Y = 4
X + 1 = 4
- 1 -1X = 3
x + y = 43 + 1 = 4 4 = 4
2x + 3y = 9
2(3) + 3(1) = 9
6 + 3 = 9
9 = 9