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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

-65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r
WENDY Reply
write in this form a/b answer should be in the simplest form 5%
August Reply
convert to decimal 9/11
August
Equation in the form of a pending point y+2=1/6(×-4)
Jose Reply
write in simplest form 3 4/2
August
definition of quadratic formula
Ahmed Reply
From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots.
Melissa
what is the answer of w-2.6=7.55
What Reply
10.15
Michael
w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6)
Korin
Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?
Mckenzie Reply
x > $110,000
bruce
greater than $110,000
Michael
Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost $1.80 per pound, and blueberries cost $4.50 per pound. If Estelle wants the fruit salad to cost her $2.52 per pound, how many pounds of each berry should she use?
nawal Reply
$1.38 worth of strawberries + $1.14 worth of blueberries which= $2.52
Leitha
how
Zaione
is it right😊
Leitha
lol maybe
Robinson
8 pound of blueberries and 22 pounds of strawberries
Melissa
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound
Melissa
hmmmm...... ?
Robinson
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
The question asks how many pounds of each in order for her to have an average cost of $2.52. She needs 30 lb in all so 30 pounds times $2.52 equals $75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60
Melissa
good
Robinson
👍
Leitha
thanks Melissa.
Leitha
nawal let's do another😊
Leitha
we can't use emojis...I see now
Leitha
Sorry for the multi post. My phone glitches.
Melissa
Vina has $4.70 in quarters, dimes and nickels in her purse. She has eight more dimes than quarters and six more nickels than quarters. How many of each coin does she have?
Mckenzie Reply
10 quarters 16 dimes 12 nickels
Leitha
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet.
Crispy Reply
wtf. is a tail wind or headwind?
Robert
48 miles per hour with headwind and 68 miles per hour with tailwind
Leitha
average speed is 58 mph
Leitha
Into the wind (headwind), 125 mph; with wind (tailwind), 175 mph. Use time (t) = distance (d) ÷ rate (r). since t is equal both problems, then 1210/(x-25) = 1694/(×+25). solve for x gives x=150.
bruce
the jet will fly 9.68 hours to cover either distance
bruce
Riley is planning to plant a lawn in his yard. He will need 9 pounds of grass seed. He wants to mix Bermuda seed that costs $4.80 per pound with Fescue seed that costs $3.50 per pound. How much of each seed should he buy so that the overall cost will be $4.02 per pound?
Vonna Reply
33.336
Robinson
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost $20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot?
Imaan Reply
Ivan has $8.75 in nickels and quarters in his desk drawer. The number of nickels is twice the number of quarters. How many coins of each type does he have?
mikayla Reply
2q=n ((2q).05) + ((q).25) = 8.75 .1q + .25q = 8.75 .35q = 8.75 q = 25 quarters 2(q) 2 (25) = 50 nickles Answer check 25 x .25 = 6.25 50 x .05 = 2.50 6.25 + 2.50 = 8.75
Melissa
John has $175 in $5 and $10 bills in his drawer. The number of $5 bills is three times the number of $10 bills. How many of each are in the drawer?
mikayla Reply
7-$10 21-$5
Robert
Enrique borrowed $23,500 to buy a car. He pays his uncle 2% interest on the $4,500 he borrowed from him, and he pays the bank 11.5% interest on the rest. What average interest rate does he pay on the total $23,500? (Round your answer to the nearest tenth of a percent.)
Parker 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 hour longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Markeice Reply
8mph
michele
16mph
Robert
3.8 mph
Ped
16 goes into 80 5times while 20 goes into 80 4times and is 4mph faster
Robert
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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