Greetings, may we use systems of equations to solve real world problems?

Yes - the one that used to be used a lot was to compare cell phone plans, but now that many are unlimited data, it is not the same. Economics uses it for example in profit loss graphs. Engineers still use systems. It is used for comparing mortgages and other comparisons.

can we find a systeme of equation that has infinite solution apply in the real life ??

Gwyndolin wants to buy some tomatoes and two vegetation vendors happen to be selling tomatoes. Stall A sells them for $0.50 each and stall B sells them for 2 for a $1.00. both stalls charge a $0.10 visit fee. stall A) y = 0.50x + 0.10 stall B) y = 1/2x + 0.10

How do u know where to put the lines?

For each equation, you need to find at least two points. Then you draw a line that goes through the points. The easiest way to find the points is to just pick a value for one of the variables (x or y) and then solve for the other variable. For instance, take the equation *y = -2x + 4* _Let's say x = 0_ y = -2x + 4 y = -2(0) + 4 y = 0 + 4 y = 4 _So when x = 0, y = 4_ Our first point is (0, 4) _Now, let's say x = 2_ y = -2x + 4 y = -2(2) + 4 y = -4 + 4 y = 0 _So when x = 2, y = 0_ Our second point is (2, 0) _Finally, let's say x = 1_ y = -2x + 4 y = -2(1) + 4 y = -2 + 4 y = 2 _So when x = 1, y = 2_ Our third point is (1, 2) Now, just graph these three points on paper (0, 4) (2, 0) (1, 2) and then draw a line that goes through all of the points. For the second equation, repeat the process to find the points and then graph the points / line on the _same_ xy graph as the first equation. You will know where to put the lines by finding the points for each equation. Hope this helps!

what is (y1-y)=m(x1-x)

That is the point slope form of a linear equation multiplied by -1. Point slope form is y-y1=m(x-x1)

What about non-linear systems of equations like 2x^2+3x-4 -0.5x^2-x+1 Could they have exactly 2 solutions?

Yes! If you equate them to each other, you'll get a third quadratic in x. Then, solving that will give you the two x coordinates of their intersection, thus proving that they do indeed have two solutions. Do note that a system of two quadratics can have a single solution too. But at max, they'll have two (unique) ones.

What is algebra 1 question

If you're relating to this question; y = −6x+8 y = −3x−4 First step: Either use substitution or elimination to find where they intersect. For this, I will use substitution. First, rearrange an equation so that you can substitute a value in; y = -6x + 8 -> x = -y/6 + 4/3. From there you substitute into any of your equations and you find y afterwards. I hope this helped!

How do you find the solution to a system without using a graph? Is it even possible to do that?

Yes, watch this video https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-systems-topic/cc-8th-systems-solutions/v/examples-algebraically-analyzing-solutions-to-systems.

How would you solve: y = -2x + 1 y = 4x - 3 It doesn't make sense. Thanks!

If y = -2x + 1 and y is also 4x - 3 then -2x + 1 = 4x - 3 so -6x = -4 so x = 4/6 = 2/3 Now that you know the value of x, you can plug it back into the equations to solve for y.

Main content

Algebra basics

Course: Algebra basics > Unit 5

Lesson 4: Number of solutions to systems of equations

Number of solutions to system of equations review

Google Classroom

A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line). This article reviews all three cases.

Want to learn more about the number of solutions to systems of equations? Check out this video.

Example system with one solution

We're asked to find the number of solutions to this system of equations:

\begin{aligned} y & = - 6 x + 8 \\ 3 x + y & = - 4 \end{aligned}

Let's put them in slope-intercept form:

\begin{aligned} y & = - 6 x + 8 \\ y & = - 3 x - 4 \end{aligned}

Since the slopes are different, the lines must intersect. Here are the graphs:

Because the lines intersect at a point, there is one solution to the system of equations the lines represent.

Example system with no solution

We're asked to find the number of solutions to this system of equations:

\begin{aligned} y & = - 3 x + 9 \\ y & = - 3 x - 7 \end{aligned}

Without graphing these equations, we can observe that they both have a slope of

- 3

. This means that the lines must be parallel. And since the

y

-intercepts are different, we know the lines are not on top of each other.

There is no solution to this system of equations.

Example system with infinite solutions

We're asked to find the number of solutions to this system of equations:

\begin{aligned} - 6 x + 4 y & = 2 \\ 3 x - 2 y & = - 1 \end{aligned}

Interestingly, if we multiply the second equation by

- 2

, we get the first equation:

\begin{aligned} 3 x - 2 y & = - 1 \\ - 2 (3 x - 2 y) & = - 2 (- 1) \\ - 6 x + 4 y & = 2 \end{aligned}

In other words, the equations are equivalent and share the same graph. Any solution that works for one equation will also work for the other equation, so there are infinite solutions to the system.

Practice

Problem 1

How many solutions does the system of linear equations have?

\begin{aligned} y & = - 2 x + 4 \\ 7 y & = - 14 x + 28 \end{aligned}

Want more practice? Check out these exercises:

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Naveen Giri
Posted 7 years ago. Direct link to Naveen Giri's post “Greetings, may we use sys...”
Greetings, may we use systems of equations to solve real world problems?
Button navigates to signup pageComment on Naveen Giri's post “Greetings, may we use sys...”
(28 votes)
Answer
- David Severin
  Posted 7 years ago. Direct link to David Severin's post “Yes - the one that used t...”
  Yes - the one that used to be used a lot was to compare cell phone plans, but now that many are unlimited data, it is not the same. Economics uses it for example in profit loss graphs. Engineers still use systems. It is used for comparing mortgages and other comparisons.
  Comment on David Severin's post “Yes - the one that used t...”
  (52 votes)
Taha Anouar
Posted 7 years ago. Direct link to Taha Anouar's post “can we find a systeme of ...”
can we find a systeme of equation that has infinite solution apply in the real life ??
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(7 votes)
Answer
- Dan
  Posted 6 years ago. Direct link to Dan's post “Gwyndolin wants to buy so...”
  Gwyndolin wants to buy some tomatoes and two vegetation vendors happen to be selling tomatoes. Stall A sells them for $0.50 each and stall B sells them for 2 for a $1.00. both stalls charge a $0.10 visit fee.
  stall A) y = 0.50x + 0.10
  stall B) y = 1/2x + 0.10
  Comment on Dan's post “Gwyndolin wants to buy so...”
  (35 votes)
Samuel
Posted 4 years ago. Direct link to Samuel's post “How do u know where to pu...”
How do u know where to put the lines?
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(4 votes)
Answer
- Polina Vitić
  Posted 4 years ago. Direct link to Polina Vitić's post “For each equation, you ne...”
  For each equation, you need to find at least two points. Then you draw a line that goes through the points.
  
  The easiest way to find the points is to just pick a value for one of the variables (x or y) and then solve for the other variable.
  
  For instance, take the equation y = -2x + 4
  
  Let's say x = 0
  y = -2x + 4
  y = -2(0) + 4
  y = 0 + 4
  y = 4
  So when x = 0, y = 4
  Our first point is (0, 4)
  
  Now, let's say x = 2
  y = -2x + 4
  y = -2(2) + 4
  y = -4 + 4
  y = 0
  So when x = 2, y = 0
  Our second point is (2, 0)
  
  Finally, let's say x = 1
  y = -2x + 4
  y = -2(1) + 4
  y = -2 + 4
  y = 2
  So when x = 1, y = 2
  Our third point is (1, 2)
  
  Now, just graph these three points on paper
  (0, 4)
  (2, 0)
  (1, 2)
  and then draw a line that goes through all of the points.
  
  For the second equation, repeat the process to find the points and then graph the points / line on the same xy graph as the first equation.
  
  You will know where to put the lines by finding the points for each equation.
  
  Hope this helps!
  Button navigates to signup page
  (15 votes)
387031
Posted 6 months ago. Direct link to 387031's post “what is (y1-y)=m(x1-x)”
what is (y1-y)=m(x1-x)
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(4 votes)
Answer
- David Severin
  Posted 6 months ago. Direct link to David Severin's post “That is the point slope f...”
  That is the point slope form of a linear equation multiplied by -1. Point slope form is y-y1=m(x-x1)
  Comment on David Severin's post “That is the point slope f...”
  (9 votes)
Pedro González
Posted 5 years ago. Direct link to Pedro González's post “How do you find the solut...”
How do you find the solution to a system without using a graph? Is it even possible to do that?
Button navigates to signup pageComment on Pedro González's post “How do you find the solut...”
(2 votes)
Answer
- m971322
  Posted 5 months ago. Direct link to m971322's post “Yes, watch this video htt...”
  Yes, watch this video https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-systems-topic/cc-8th-systems-solutions/v/examples-algebraically-analyzing-solutions-to-systems.
  Button navigates to signup page
  (2 votes)
IDK TBH
Posted 3 years ago. Direct link to IDK TBH's post “How would you solve: ...”
How would you solve:
y = -2x + 1
y = 4x - 3
It doesn't make sense. Thanks!
Button navigates to signup pageComment on IDK TBH's post “How would you solve: ...”
(2 votes)
Answer
- Mariette
  Posted a year ago. Direct link to Mariette's post “If y = -2x + 1 and y is a...”
  If y = -2x + 1
  and y is also 4x - 3
  then -2x + 1 = 4x - 3
  so -6x = -4
  so x = 4/6 = 2/3
  Now that you know the value of x, you can plug it back into the equations to solve for y.
  Button navigates to signup page
  (3 votes)
Marcus king
Posted a year ago. Direct link to Marcus king's post “it the year 2023”
it the year 2023
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Answer
WarmSausageTea‭ ⛮
Posted 8 months ago. Direct link to WarmSausageTea‭ ⛮'s post “What about non-linear sys...”
What about non-linear systems of equations like
2x^2+3x-4
-0.5x^2-x+1
Could they have exactly 2 solutions?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Venkata
  Posted 8 months ago. Direct link to Venkata's post “Yes! If you equate them t...”
  Yes! If you equate them to each other, you'll get a third quadratic in x. Then, solving that will give you the two x coordinates of their intersection, thus proving that they do indeed have two solutions.
  
  Do note that a system of two quadratics can have a single solution too. But at max, they'll have two (unique) ones.
  Button navigates to signup page
  (5 votes)
msbacon700
Posted 4 years ago. Direct link to msbacon700's post “What is algebra 1 questio...”
What is algebra 1 question
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(3 votes)
Answer
- Victor
  Posted 4 years ago. Direct link to Victor's post “If you're relating to thi...”
  If you're relating to this question;
  y = −6x+8
  y = −3x−4
  First step: Either use substitution or elimination to find where they intersect. For this, I will use substitution.
  First, rearrange an equation so that you can substitute a value in; y = -6x + 8 -> x = -y/6 + 4/3. From there you substitute into any of your equations and you find y afterwards. I hope this helped!
  
  Button navigates to signup page
  (2 votes)
😊
Posted 7 years ago. Direct link to 😊's post “Can anybody explain me .....”
Can anybody explain me ...........in the 1st equation which is Y=-6x+8 ............i know it`s from y intercept chapter.....How do you draw on graph ?on this equation 6 is slope and 8 is the value of y......so 8+ goes to y+ graph but from where did you get other point ........it looks like more than 1.....How do you figure it out ?
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(4 votes)
Answer
- joshkimm2004
  Posted 7 years ago. Direct link to joshkimm2004's post “ok, so first off, the slo...”
  ok, so first off, the slope is -6 not 6 because of the negative sign in front of the -6x. And then you use the 8 as the intercept. as a result, one of the points become (0,8). Then, you go six down and one to the right. That's another point and you can keep doing that.
  Comment on joshkimm2004's post “ok, so first off, the slo...”
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