1. Home
  2. Elementary algebra
  3. Systems of linear equations
  4. Solve systems of equations by

5.1 Solve systems of equations by graphing  (Page 2/14)

<< Chapter < Page
  Elementary algebra     Page 2 / 14
Chapter >> Page >

Determine whether the ordered pair is a solution to the system: { 3 x + y = 0 x + 2 y = −5 .

ⓐ ( 1 , −3 ) ⓑ ( 0 , 0 )

ⓐ yes ⓑ no

Got questions? Get instant answers now!

Determine whether the ordered pair is a solution to the system: { x − 3 y = −8 −3 x − y = 4 .

ⓐ ( 2 , −2 ) ⓑ ( −2 , 2 )

ⓐ no ⓑ yes

Got questions? Get instant answers now!

Solve a system of linear equations by graphing

In this chapter we will use three methods to solve a system of linear equations. The first method we’ll use is graphing.

The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we’ll find the solution to the system.

Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions.

Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown in [link] :

This figure shows three x y-coordinate planes. The first plane shows two lines which intersect at one point. Under the graph it says, “The lines intersect. Intersecting lines have one point in common. There is one solution to this system.” The second x y-coordinate plane shows two parallel lines. Under the graph it says, “The lines are parallel. Parallel lines have no points in common. There is no solution to this system.” The third x y-coordinate plane shows one line. Under the graph it says, “Both equations give the same line. Because we have just one line, there are infinitely many solutions.”

For the first example of solving a system of linear equations in this section and in the next two sections, we will solve the same system of two linear equations. But we’ll use a different method in each section. After seeing the third method, you’ll decide which method was the most convenient way to solve this system.

How to solve a system of linear equations by graphing

Solve the system by graphing: { 2 x + y = 7 x − 2 y = 6 .

Solution

This table has four rows and three columns. The first column acts as the header column. The first row reads, “Step 1. Graph the first equation.” Then it reads, “To graph the first line, write the equation in slope-intercept form.” The equation reads 2x + y = 7 and becomes y = -2x + 7 where m = -2 and b = 7. Then it shows a graph of the equations 2x + y = 7. The equation x – 2y = 6 is also listed. The second row reads, “Step 2. Graph the second equation on the same rectangular coordinate system.” Then it says, “To graph the second line, use intercepts.” This is followed by the equation x – 2y = 6 and the ordered pairs (0, -3) and (6, 0). The last column of this row shows a graph of the two equations. The third row reads, “Step 3. Determine whether the lines intersect, are parallel, or are the same line.” Then “Look at the graph of the lines.” Finally it reads, “The lines intersect.” The fourth row reads, “Step 4. Identify the solution to the system. If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions.” Then it reads, “Since the lines intersect, find the point of intersection. Check the point in both equations.” Finally it reads, “The lines intersect at (4, -1). It then uses substitution to show that, “The solution is (4, -1).”
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve each system by graphing: { x − 3 y = −3 x + y = 5 .

( 3 , 2 )

Got questions? Get instant answers now!

Solve each system by graphing: { − x + y = 1 3 x + 2 y = 12 .

( 2 , 3 )

Got questions? Get instant answers now!

The steps to use to solve a system of linear equations by graphing are shown below.

To solve a system of linear equations by graphing.

  1. Graph the first equation.
  2. Graph the second equation on the same rectangular coordinate system.
  3. Determine whether the lines intersect, are parallel, or are the same line.
  4. Identify the solution to the system.
    • If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system.
    • If the lines are parallel, the system has no solution.
    • If the lines are the same, the system has an infinite number of solutions.

Solve the system by graphing: { y = 2 x + 1 y = 4 x − 1 .

Solution

Both of the equations in this system are in slope-intercept form, so we will use their slopes and y -intercepts to graph them. { y = 2 x + 1 y = 4 x − 1

Find the slope and y -intercept of the
first equation.		 .
Find the slope and y -intercept of the
first equation.		 .
Graph the two lines.
Determine the point of intersection. The lines intersect at (1, 3).
.
Check the solution in both equations. y = 2 x + 1 3 = ? 2 · 1 + 1 3 = 3 ✓ y = 4 x − 1 3 = ? 4 · 1 − 1 3 = 3 ✓
The solution is (1, 3).

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve each system by graphing: { y = 2 x + 2 y = − x − 4 .

( −2 , −2 )

Got questions? Get instant answers now!

Solve each system by graphing: { y = 3 x + 3 y = − x + 7 .

( 1 , 6 )

Got questions? Get instant answers now!

Both equations in [link] were given in slope–intercept form. This made it easy for us to quickly graph the lines. In the next example, we’ll first re-write the equations into slope–intercept form.

Solve the system by graphing: { 3 x + y = −1 2 x + y = 0 .

Solution

We’ll solve both of these equations for y so that we can easily graph them using their slopes and y -intercepts. { 3 x + y = −1 2 x + y = 0

Solve the first equation for y .


Find the slope and y -intercept.


Solve the second equation for y .


Find the slope and y -intercept.		 3 x + y = − 1 y = − 3 x − 1 m = − 3 b = − 1 2 x + y = 0 y = − 2 x m = − 2 b = 0
Graph the lines. .
Determine the point of intersection. The lines intersect at (−1, 2).
Check the solution in both equations. 3 x + y = − 1 3 ( − 1 ) + 2 = ? − 1 − 1 = − 1 ✓ 2 x + y = 0 2 ( − 1 ) + 2 = ? 0 0 = 0 ✓
The solution is (−1, 2).

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

prostaglandin and fever
Maha Reply
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
John Reply
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Ummi Reply
omeprazole
Kamara
what
Renee
what is this
Renee
is a drug
Kamara
of anti-ulcer
Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
Nency Reply
Not really sure
Eli
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
what is anatomy
Oyindarmola Reply
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Oyerinde Reply
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Mohammed Reply
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
Wulku Reply
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
Ali Reply
function of digestive
Ali
the diagram of the lungs
Adaeze Reply
what is the normal body temperature
Diya Reply
37 degrees selcius
Xolo
37°c
Stephanie
please why 37 degree selcius normal temperature
Mark
36.5
Simon
37°c
Iyogho
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
37A c
Wulku
what is anaemia
Diya Reply
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
Diya Reply
how does Lysin attack pathogens
Diya
acid
Mary
I information on anatomy position and digestive system and there enzyme
Elisha Reply
anatomy of the female external genitalia
Muhammad Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 7

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask