4.1 Use the rectangular coordinate system  (Page 4/12)

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Which of the following ordered pairs are solutions to the equation $y=4x-3$ ?

$\left(0,3\right)$ $\left(1,1\right)$ $\left(-1,-1\right)$

b

Which of the following ordered pairs are solutions to the equation $y=-2x+6$ ?

$\left(0,6\right)$ $\left(1,4\right)$ $\left(-2,-2\right)$

a, b

Complete a table of solutions to a linear equation in two variables

In the examples above, we substituted the x - and y -values of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do you find the ordered pairs if they are not given? It’s easier than you might think—you can just pick a value for $x$ and then solve the equation for $y$ . Or, pick a value for $y$ and then solve for $x$ .

We’ll start by looking at the solutions to the equation $y=5x-1$ that we found in [link] . We can summarize this information in a table of solutions, as shown in [link] .

 $y=5x-1$ $x$ $y$ $\left(x,y\right)$ 0 $-1$ $\left(0,-1\right)$ 1 4 $\left(1,4\right)$

To find a third solution, we’ll let $x=2$ and solve for $y$ .

The ordered pair $\left(2,9\right)$ is a solution to $y=5x-1$ . We will add it to [link] .

 $y=5x-1$ $x$ $y$ $\left(x,y\right)$ 0 $-1$ $\left(0,-1\right)$ 1 4 $\left(1,4\right)$ 2 9 $\left(2,9\right)$

We can find more solutions to the equation by substituting in any value of $x$ or any value of $y$ and solving the resulting equation to get another ordered pair that is a solution. There are infinitely many solutions of this equation.

Complete [link] to find three solutions to the equation $y=4x-2$ .

 $y=4x-2$ $x$ $y$ $\left(x,y\right)$ 0 $-1$ 2

Solution

Substitute $x=0$ , $x=-1$ , and $x=2$ into $y=4x-2$ .

The results are summarized in [link] .

 $y=4x-2$ $x$ $y$ $\left(x,y\right)$ 0 $-2$ $\left(0,-2\right)$ $-1$ $-6$ $\left(-1,-6\right)$ 2 6 $\left(2,6\right)$

Complete the table to find three solutions to this equation: $y=3x-1$ .

 $y=3x-1$ $x$ $y$ $\left(x,y\right)$ 0 $-1$ 2
 $y=3x-1$ $x$ $y$ $\left(x,y\right)$ 0 $-1$ $\left(0,-1\right)$ $-1$ $-4$ $\left(-1,-4\right)$ 2 5 $\left(2,5\right)$

Complete the table to find three solutions to this equation: $y=6x+1$ .

 $y=6x+1$ $x$ $y$ $\left(x,y\right)$ 0 1 $-2$
 $y=6x+1$ $x$ $y$ $\left(x,y\right)$ 0 1 $\left(0,1\right)$ 1 7 $\left(1,7\right)$ $-2$ $-11$ $\left(-2,-11\right)$

Complete [link] to find three solutions to the equation $5x-4y=20$ .

 $5x-4y=20$ $x$ $y$ $\left(x,y\right)$ 0 0 5

Solution

Substitute the given value into the equation $5x-4y=20$ and solve for the other variable. Then, fill in the values in the table.

The results are summarized in [link] .

 $5x-4y=20$ $x$ $y$ $\left(x,y\right)$ 0 $-5$ $\left(0,-5\right)$ 4 0 $\left(4,0\right)$ 8 5 $\left(8,5\right)$

Complete the table to find three solutions to this equation: $2x-5y=20$ .

 $2x-5y=20$ $x$ $y$ $\left(x,y\right)$ 0 0 $-5$
 $2x-5y=20$ $x$ $y$ $\left(x,y\right)$ 0 $-4$ $\left(0,-4\right)$ 10 0 $\left(10,0\right)$ $-5$ $-6$ $\left(-5,-6\right)$

Complete the table to find three solutions to this equation: $3x-4y=12$ .

 $3x-4y=12$ $x$ $y$ $\left(x,y\right)$ 0 0 $-4$
 $3x-4y=12$ $x$ $y$ $\left(x,y\right)$ 0 $-3$ $\left(0,-3\right)$ 4 0 $\left(4,0\right)$ $-4$ $-6$ $\left(-4,-6\right)$

Find solutions to a linear equation

To find a solution to a linear equation, you really can pick any number you want to substitute into the equation for $x$ or $y.$ But since you’ll need to use that number to solve for the other variable it’s a good idea to choose a number that’s easy to work with.

When the equation is in y -form, with the y by itself on one side of the equation, it is usually easier to choose values of $x$ and then solve for $y$ .

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

Solution

We can substitute any value we want for $x$ or any value for $y$ . Since the equation is in y -form, it will be easier to substitute in values of $x$ . Let’s pick $x=0$ , $x=1$ , and $x=-1$ .

 Substitute the value into the equation. Simplify. Simplify. Write the ordered pair. (0, 2) (1, −1) (−1, 5) Check. $\phantom{\rule{0.03em}{0ex}}y=-3x+2$ $\phantom{\rule{0.7em}{0ex}}y=-3x+2$ $\phantom{\rule{0.04em}{0ex}}y=-3x+2$ $2\stackrel{?}{=}-3\cdot 0+2$ $-1\stackrel{?}{=}-3\cdot 1+2$ $5\stackrel{?}{=}-3\left(-1\right)+2$ $2\stackrel{?}{=}0+2$ $-1\stackrel{?}{=}-3+2$ $5\stackrel{?}{=}3+2$ $2=2✓$ $-1=-1✓$ $5=5✓$

So, $\left(0,2\right)$ , $\left(1,-1\right)$ and $\left(-1,5\right)$ are all solutions to $y=-3x+2$ . We show them in [link] .

 $y=-3x+2$ $x$ $y$ $\left(x,y\right)$ 0 2 $\left(0,2\right)$ 1 $-1$ $\left(1,-1\right)$ $-1$ 5 $\left(-1,5\right)$

Questions & Answers

where's the answers?
Ed Reply
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
Maddy Reply
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
Florence Reply
What does 3 times your weight right now
Cherokee Reply
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul
What is the lcm of 340
Kendra Reply
Yes
Cherokee
How many numbers each equal to y must be taken to make 15xy
Malik Reply
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
Adda Reply
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Adda
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
Adda
(×+1)(×-4) = x^2-3×-4
Gee
Quadratic equations involving factorization
Winner Reply
there's a chatting option in the app wow
Nana
That's cool cool
Nana
Nice to meet you all
Nana
you too.
Joan
😃
Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
me too as part of my core subjects in R M E
Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
vinaya Reply
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs$4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon? Mohamed Reply (a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r) muhammad Reply 4x-7y=8 2x-7y=1 what is the answer? Ramil Reply x=7/2 & y=6/7 Pbp x=7/2 & y=6/7 use Elimination Debra true bismark factoriz e usman 4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2 Peggie Ok cool answer peggie Frank thanks Ramil copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x Sandra Reply What is c+4=8 Penny Reply 2 Letha 4 Lolita 4 Rich 4 thinking C+4=8 -4 -4 C =4 thinking I need to study Letha 4+4=8 William During two years in college, a student earned$9,500. The second year, she earned \$500 more than twice the amount she earned the first year.
Nicole Reply
9500=500+2x
Debra
9500-500=9000 9000÷2×=4500 X=4500
Debra
X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500
Pbp

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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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