# 4.2 Graph linear equations in two variables

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By the end of this section, you will be able to:
• Recognize the relationship between the solutions of an equation and its graph.
• Graph a linear equation by plotting points.
• Graph vertical and horizontal lines.

Before you get started, take this readiness quiz.

1. Evaluate $3x+2$ when $x=-1$ .
If you missed this problem, review [link] .
2. Solve $3x+2y=12$ for $y$ in general.
If you missed this problem, review [link] .

## Recognize the relationship between the solutions of an equation and its graph

In the previous section, we found several solutions to the equation $3x+2y=6$ . They are listed in [link] . So, the ordered pairs $\left(0,3\right)$ , $\left(2,0\right)$ , and $\left(1,\frac{3}{2}\right)$ are some solutions to the equation $3x+2y=6$ . We can plot these solutions in the rectangular coordinate system as shown in [link] .

 $3x+2y=6$ $x$ $y$ $\left(x,y\right)$ 0 3 $\left(0,3\right)$ 2 0 $\left(2,0\right)$ 1 $\frac{3}{2}$ $\left(1,\frac{3}{2}\right)$

Notice how the points line up perfectly? We connect the points with a line to get the graph of the equation $3x+2y=6$ . See [link] . Notice the arrows on the ends of each side of the line. These arrows indicate the line continues.

Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are not solutions.

Notice that the point whose coordinates are $\left(-2,6\right)$ is on the line shown in [link] . If you substitute $x=-2$ and $y=6$ into the equation, you find that it is a solution to the equation. So the point $\left(-2,6\right)$ is a solution to the equation $3x+2y=6$ . (The phrase “the point whose coordinates are $\left(-2,6\right)$ ” is often shortened to “the point $\left(-2,6\right)$ .”) So $\left(4,1\right)$ is not a solution to the equation $3x+2y=6$ . Therefore, the point $\left(4,1\right)$ is not on the line. See [link] . This is an example of the saying, “A picture is worth a thousand words.” The line shows you all the solutions to the equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation $3x+2y=6$ .

## Graph of a linear equation

The graph of a linear equation     $Ax+By=C$ is a line.

• Every point on the line is a solution of the equation.
• Every solution of this equation is a point on this line.

The graph of $y=2x-3$ is shown. For each ordered pair, decide:

Is the ordered pair a solution to the equation?
Is the point on the line?

A $\left(0,-3\right)$  B $\left(3,3\right)$  C $\left(2,-3\right)$  D $\left(-1,-5\right)$

## Solution

Substitute the x - and y - values into the equation to check if the ordered pair is a solution to the equation.

1. 2. Plot the points A $\left(0,3\right)$ , B $\left(3,3\right)$ , C $\left(2,-3\right)$ , and D $\left(-1,-5\right)$ . The points $\left(0,3\right)$ , $\left(3,3\right)$ , and $\left(-1,-5\right)$ are on the line $y=2x-3$ , and the point $\left(2,-3\right)$ is not on the line.

The points that are solutions to $y=2x-3$ are on the line, but the point that is not a solution is not on the line.

Use the graph of $y=3x-1$ to decide whether each ordered pair is:

• a solution to the equation.
• on the line.

$\left(0,-1\right)$ $\left(2,5\right)$ yes, yes  yes, yes

Use graph of $y=3x-1$ to decide whether each ordered pair is:

• a solution to the equation
• on the line

$\left(3,-1\right)$ $\left(-1,-4\right)$ no, no  yes, yes

## Graph a linear equation by plotting points

There are several methods that can be used to graph a linear equation. The method we used to graph $3x+2y=6$ is called plotting points, or the Point–Plotting Method.

#### Questions & Answers

Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was$1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was$1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
Amara currently sells televisions for company A at a salary of $17,000 plus a$100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a$20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
gustavo Reply
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
Wenda Reply
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned$1600 more than Jim earned. How much did they earned
Arleathia Reply
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
Sunnyshay Reply
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike
Washing his dad’s car alone, eight year old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Gagan Reply
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
1,75 hrs is correct Mike
Emund
How did you come up with the answer?
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Marie Reply
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have? Kendra Reply Tickets to a Broadway show cost$35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were$47,150. How many adult and how many child tickets were sold?
dana Reply
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received$4,500 in interest in one year? How do I do this
Tanesia Reply
how to square
Fiona Reply
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
Sabee Reply

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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