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Find three solutions to this equation: y = −2 x + 3 .

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Find three solutions to this equation: y = −4 x + 1 .

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We have seen how using zero as one value of x makes finding the value of y easy. When an equation is in standard form, with both the x and y on the same side of the equation, it is usually easier to first find one solution when x = 0 find a second solution when y = 0 , and then find a third solution.

Find three solutions to the equation 3 x + 2 y = 6 .

Solution

We can substitute any value we want for x or any value for y . Since the equation is in standard form, let’s pick first x = 0 , then y = 0 , and then find a third point.

. . .
. . .
Substitute the value into the equation. . . .
Simplify. . . .
Solve. . . .
. . .
Write the ordered pair. (0, 3) (2, 0) ( 1 , 3 2 )
Check.
3 x + 2 y = 6 3 x + 2 y = 6 3 x + 2 y = 6
3 0 + 2 3 6 3 2 + 2 0 6 3 1 + 2 3 2 6
0 + 6 6 6 + 0 6 3 + 3 6
6 = 6 6 = 6 6 = 6

So ( 0 , 3 ) , ( 2 , 0 ) , and ( 1 , 3 2 ) are all solutions to the equation 3 x + 2 y = 6 . We can list these three solutions in [link] .

3 x + 2 y = 6
x y ( x , y )
0 3 ( 0 , 3 )
2 0 ( 2 , 0 )
1 3 2 ( 1 , 3 2 )

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Find three solutions to the equation 2 x + 3 y = 6 .

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Find three solutions to the equation 4 x + 2 y = 8 .

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Key concepts

  • Sign Patterns of the Quadrants
    Quadrant I Quadrant II Quadrant III Quadrant IV ( x , y ) ( x , y ) ( x , y ) ( x , y ) ( + , + ) ( , + ) ( , ) ( + , )
  • Points on the Axes
    • On the x -axis, y = 0 . Points with a y -coordinate equal to 0 are on the x -axis, and have coordinates ( a , 0 ) .
    • On the y -axis, x = 0 . Points with an x -coordinate equal to 0 are on the y -axis, and have coordinates ( 0 , b ) .
  • Solution of a Linear Equation
    • An ordered pair ( x , y ) is a solution of the linear equation A x + B y = C , if the equation is a true statement when the x - and y - values of the ordered pair are substituted into the equation.

Practice makes perfect

Plot Points in a Rectangular Coordinate System

In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.


( −4 , 2 )
( −1 , −2 )
( 3 , −5 )
( −3 , 5 )
( 5 3 , 2 )

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 4, 2) is plotted and labeled

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( −2 , −3 )
( 3 , −3 )
( −4 , 1 )
( 4 , −1 )
( 3 2 , 1 )

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( 3 , −1 )
( −3 , 1 )
( −2 , 2 )
( −4 , −3 )
( 1 , 14 5 )

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (3, negative 1) is plotted and labeled

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( −1 , 1 )
( −2 , −1 )
( 2 , 1 )
( 1 , −4 )
( 3 , 7 2 )

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In the following exercises, plot each point in a rectangular coordinate system.


( −2 , 0 )
( −3 , 0 )
( 0 , 0 )
( 0 , 4 )
( 0 , 2 )

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 2, 0) is plotted and labeled

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( 0 , 1 )
( 0 , −4 )
( −1 , 0 )
( 0 , 0 )
( 5 , 0 )

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( 0 , 0 )
( 0 , −3 )
( −4 , 0 )
( 1 , 0 )
( 0 , −2 )

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, 0) is plotted and labeled

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( −3 , 0 )
( 0 , 5 )
( 0 , −2 )
( 2 , 0 )
( 0 , 0 )

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In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 4, 1) is plotted and labeled “A”. The point (negative 3, negative 4) is plotted and labeled “B”. The point (1, negative 3) is plotted and labeled “C”. The point (4, 3) is plotted and labeled “D”.

A: ( −4 , 1 )  B: ( −3 , −4 )  C: ( 1 , −3 )  D: ( 4 , 3 )

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The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, negative 2) is plotted and labeled “A”. The point (negative 2, 0) is plotted and labeled “B”. The point (0, 5) is plotted and labeled “C”. The point (5, 0) is plotted and labeled “D”.

A: ( 0 , −2 )  B: ( −2 , 0 )  C: ( 0 , 5 )  D: ( 5 , 0 )

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Verify Solutions to an Equation in Two Variables

In the following exercises, which ordered pairs are solutions to the given equations?

2 x + y = 6

( 1 , 4 )
( 3 , 0 )
( 2 , 3 )

a, b

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x + 3 y = 9

( 0 , 3 )
( 6 , 1 )
( −3 , −3 )

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4 x 2 y = 8

( 3 , 2 )
( 1 , 4 )
( 0 , −4 )

a, c

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3 x 2 y = 12

( 4 , 0 )
( 2 , −3 )
( 1 , 6 )

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y = 4 x + 3

( 4 , 3 )
( −1 , −1 )
( 1 2 , 5 )

b, c

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y = 2 x 5

( 0 , −5 )
( 2 , 1 )
( 1 2 , −4 )

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y = 1 2 x 1

( 2 , 0 )
( −6 , −4 )
( −4 , −1 )

a, b

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y = 1 3 x + 1

( −3 , 0 )
( 9 , 4 )
( −6 , −1 )

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Complete a Table of Solutions to a Linear Equation

In the following exercises, complete the table to find solutions to each linear equation.

y = 2 x 4

x y ( x , y )
0
2
−1
x y ( x , y )
0 −4 ( 0 , −4 )
2 0 ( 2 , 0 )
−1 −6 ( −1 , −6 )
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y = 3 x 1

x y ( x , y )
0
2
−1
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y = x + 5

x y ( x , y )
0
3
−2
x y ( x , y )
0 5 ( 0 , 5 )
3 2 ( 3 , 2 )
−2 7 ( −2 , 7 )
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y = x + 2

x y ( x , y )
0
3
−2
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y = 1 3 x + 1

x y ( x , y )
0
3
6
x y ( x , y )
0 1 ( 0 , 1 )
3 2 ( 3 , 2 )
6 3 ( 6 , 3 )
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y = 1 2 x + 4

x y ( x , y )
0
2
4
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y = 3 2 x 2

x y ( x , y )
0
2
−2
x y ( x , y )
0 −2 ( 0 , −2 )
2 −5 ( 2 , −5 )
−2 1 ( −2 , 1 )
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y = 2 3 x 1

x y ( x , y )
0
3
−3
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x + 3 y = 6

x y ( x , y )
0
3
0
x y ( x , y )
0 2 ( 0 , 2 )
3 4 ( 3 , 1 )
6 0 ( 6 , 0 )
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x + 2 y = 8

x y ( x , y )
0
4
0
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2 x 5 y = 10

x y ( x , y )
0
10
0
x y ( x , y )
0 −2 ( 0 , −2 )
10 2 ( 10 , 2 )
5 0 ( 5 , 0 )
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3 x 4 y = 12

x y ( x , y )
0
8
0
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Find Solutions to a Linear Equation

In the following exercises, find three solutions to each linear equation.

y = 5 x 8

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y = −4 x + 5

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x + y = 8

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x + y = −2

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3 x + y = 5

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4 x y = 8

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2 x + 4 y = 8

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5 x 2 y = 10

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Everyday math

Weight of a baby. Mackenzie recorded her baby’s weight every two months. The baby’s age, in months, and weight, in pounds, are listed in the table below, and shown as an ordered pair in the third column.

Plot the points on a coordinate plane.

.

Why is only Quadrant I needed?

Age x Weight y ( x , y )
0 7 (0, 7)
2 11 (2, 11)
4 15 (4, 15)
6 16 (6, 16)
8 19 (8, 19)
10 20 (10, 20)
12 21 (12, 21)


The graph shows the x y-coordinate plane. The x- and y-axes each run from 0 to 25. The points (0, 7), (2, 11), (4, 15), (6, 16), (8, 19), (10, 20) and (12, 21) are plotted and labeled.
Age and weight are only positive.

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Weight of a child. Latresha recorded her son’s height and weight every year. His height, in inches, and weight, in pounds, are listed in the table below, and shown as an ordered pair in the third column.

Plot the points on a coordinate plane.

.

Why is only Quadrant I needed?

Height x Weight y ( x , y )
28 22 (28, 22)
31 27 (31, 27)
33 33 (33, 33)
37 35 (37, 35)
40 41 (40, 41)
42 45 (42, 45)
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Writing exercises

Explain in words how you plot the point ( 4 , −2 ) in a rectangular coordinate system.

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How do you determine if an ordered pair is a solution to a given equation?

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Is the point ( −3 , 0 ) on the x -axis or y -axis? How do you know?

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Is the point ( 0 , 8 ) on the x -axis or y -axis? How do you know?

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “plot points in a rectangular coordinate system,”, “identify points on a graph,” “verify solutions to an equation in two variables,” “complete a table of solutions to a linear equation,” and “find solutions to a linear equation.” The rest of the cells are blank.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no, I don’t get it. This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

Questions & Answers

how many typos can we find...?
Chris Reply
5
Joseph
In the LCM Prime Factors exercises, the LCM of 28 and 40 is 280. Not 420!
Chris Reply
4x+7y=29,x+3y=11 substitute method of linear equation
Srinu Reply
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce
At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
Andrew Reply
divide 3x⁴-4x³-3x-1 by x-3
Ritik Reply
how to multiply the monomial
Ceny Reply
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
Seera Reply
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Juned Reply
Brandon has a cup of quarters and dimes with a total of 5.55$. The number of quarters is five less than three times the number of dimes
ashley Reply
app is wrong how can 350 be divisible by 3.
Raheem Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna Reply
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
Lorris Reply
I'm getting "math processing error" on math problems. Anyone know why?
Ray Reply
Can you all help me I don't get any of this
Jade Reply
4^×=9
Alberto Reply
Practice Key Terms 7

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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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