· identify whether a device of linear equations is continuous or inconsistent.

You are watching: How many solutions are there for the system of equations shown on the graph?

· identify whether a system of linear equations is dependency or independent.

· determine whether an ordered pair is a solution of a device of equations.

· solve application problems by graphing a system of equations.


Recall the a straight equation graphs together a line, which suggests that every one of the points on the heat are remedies to that straight equation. There are an infinite number of solutions. If you have actually a system of straight equations, the solution for the system is the value that makes all of the equations true. For 2 variables and also two equations, this is the point where the two graphs intersect. The collaborates of this suggest will it is in the equipment for the 2 variables in the two equations.


The solution for a device of equations is the worth or worths that space true for every equations in the system. The graphs that equations within a system deserve to tell girlfriend how countless solutions exist for that system. Look at the photos below. Each reflects two present that comprise a mechanism of equations.

One Solution

No Solutions

Infinite Solutions

*

*

*

If the graphs that the equations intersect, climate there is one systems that is true for both equations.

If the graphs that the equations do not intersect (for example, if they space parallel), then there room no solutions that space true because that both equations.

If the graphs of the equations are the same, climate there are an infinite variety of solutions that are true for both equations.

When the lines intersect, the suggest of intersection is the only point that the two graphs have in common. For this reason the coordinates of that point are the solution for the two variables provided in the equations. When the lines room parallel, there room no solutions, and sometimes the 2 equations will graph as the exact same line, in which situation we have an infinite number of solutions.

Some one-of-a-kind terms are sometimes used to define these kinds of systems.

The complying with terms refer to how numerous solutions the mechanism has.

o as soon as a system has actually one solution (the graphs the the equations intersect once), the system is a consistent mechanism of straight equations and the equations space independent.

o once a system has actually no solution (the graphs of the equations don’t intersect at all), the device is an inconsistent system of linear equations and the equations room independent.

o If the lines space the very same (the graphs crossing at all points), the system is a constant system of direct equations and the equations room dependent. The is, any solution that one equation must additionally be a solution of the other, therefore the equations count on every other.

The adhering to terms refer to whether the system has any type of solutions in ~ all.

o The system is a constant system of linear equations once it has solutions.

o The mechanism is one inconsistent system of linear equations when it has actually no solutions.

We can summarize this together follows:

o A mechanism with one or much more solutions is consistent.

o A system with no solutions is inconsistent.

o If the lines space different, the equations room independent direct equations.

o If the lines are the same, the equations are dependent straight equations.


Example

Problem

Using the graph that y = x and also x + 2y = 6, presented below, identify how many solutions the mechanism has. Then classify the mechanism as constant or inconsistent and the equations as dependent or independent.

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The lines crossing at one point. Therefore the 2 lines have only one point in common, over there is only one equipment to the system.

Because the lines space not the exact same the equations space independent.

Because over there is just one solution, this system is consistent.

Answer

The system is consistent and the equations are independent.


Advanced Example

Problem

Using the graph the y = 3.5x + 0.25 and also 14x – 4y = -4.5, presented below, identify how countless solutions the mechanism has. Then classify the system as continual or inconsistent and the equations as dependent or independent.

*

The lines room parallel, definition they execute not intersect. There room no remedies to the system.

The lines room not the same, the equations are independent.

There are no solutions. Therefore, this device is inconsistent.

Answer

The mechanism is inconsistent and the equations are independent.


Advanced Question

Which of the complying with represents dependency equations and also consistent systems?

A)

B)

C)

D)


A)

Incorrect. The 2 lines in this system have the very same slope, however different worths for b. This method the lines are parallel. The lines don’t intersect, for this reason there room no solutions and the mechanism is inconsistent. Since the lines are not the very same the equations are independent. The correct answer is C.

B)

Incorrect. The two lines in this mechanism have different slopes and different values for b. This method the lines crossing at one point. Due to the fact that there is a solution, this device is consistent. And because the lines are not the same, the equations space independent. The correct answer is C.

C)

Correct. The two lines in this device are the same;  can it is in rewritten together . Since there are numerous solutions, this system is consistent. The present are identical so the equations space dependent.

D)

Incorrect. The 2 lines in this system have various slopes and the exact same value for b. This means the lines intersect at one point—the y-intercept. Recall the intersecting lines have one solution and therefore the mechanism is consistent. Because the lines space not the very same the equations are independent. The correct answer is C.

From the graph above, you can see that there is one systems to the system y = x and also x + 2y = 6. The solution appears to it is in (2, 2). However, you should verify an answer that you review from a graph come be sure that it’s no really (2.001, 2.001) or (1.9943, 1.9943).

One way of verifying the the allude does exist ~ above both present is to substitute the x- and also y-values of the ordered pair right into the equation of every line. If the substitution results in a true statement, climate you have the correct solution!


Example

Problem

Is (2, 2) a solution of the mechanism y = x and x + 2y = 6?

y = x

2 = 2

TRUE

(2, 2) is a equipment of y = x.

x + 2y = 6

2 + 2(2) = 6

2 + 4 = 6

6 = 6

TRUE

(2, 2) is a solution of x + 2y = 6.

Since the equipment of the device must be a solution to all the equations in the system, check the allude in every equation. Substitute 2 because that x and 2 because that y in every equation.

Answer

(2, 2) is a systems to the system.

Since (2, 2) is a solution of each of the equations in the system, (2, 2) is a equipment of the system.


Example

Problem

Is (3, 9) a systems of the system y = 3x and also 2x – y = 6?

y = 3x

9 = 3(3)

TRUE

(3, 9) is a equipment of y = 3x.

2x – y = 6

2(3) – 9 = 6

6 – 9 = 6

-3 = 6

FALSE

(3, 9) is no a equipment of 2x – y = 6.

Since the solution of the system must it is in a systems to all the equations in the system, inspect the suggest in every equation. Substitute 3 because that x and 9 because that y in every equation.

Answer

(3, 9) is not a equipment to the system.

Since (3, 9) is not a systems of among the equations in the system, it can not be a solution of the system.


Example

Problem

Is (−2, 4) a systems of the mechanism y = 2x and 3x + 2y = 1?

y = 2x

4 = 2(−2)

4 = −4

FALSE

(−2, 4) is no a systems of y = 2x.

3x + 2y = 1

3(−2) + 2(4) = 1

−6 + 8 = 1

2 = 1

FALSE

(−2, 4) is not a equipment of 3x + 2y = 1.

Since the equipment of the device must be a equipment to all the equations in the system, check the allude in each equation. Instead of −2 because that x and also 4 because that y in every equation.

Answer

(−2, 4) is not a solution to the system.

Since (−2, 4) is no a systems to one of two people of the equations in the system, (−2, 4) is not a systems of the system.


Remember, the in bespeak to it is in a solution to the mechanism of equations, the worth of the point must it is in a systems for both equations. When you find one equation for which the point is false, you have established that that is no a systems for the system.

Which that the following statements is true for the system 2x – y = −3 and y = 4x – 1?

A) (2, 7) is a systems of one equation yet not the other, so it is a solution of the system

B) (2, 7) is a equipment of one equation however not the other, so the is no a solution of the system

C) (2, 7) is a solution of both equations, so that is a equipment of the system

D) (2, 7) is no a equipment of either equation, so that is no a systems to the system


A) (2, 7) is a equipment of one equation but not the other, so the is a equipment of the system

Incorrect. If the allude were a equipment of one equation however not the other, then it is no a equipment of the system. In fact, the point (2, 7) is a equipment of both equations, so it is a equipment of the system. The 2 lines are not identical, so the is the only solution.

B) (2, 7) is a systems of one equation but not the other, so the is no a equipment of the system

Incorrect. The suggest (2, 7) is a systems of both equations, so it is a solution of the system. The 2 lines room not identical, so that is the only solution.

C) (2, 7) is a solution of both equations, so that is a solution of the mechanism

Correct. Substituting 2 for x and also 7 for y gives true declaration in both equations, so the allude is a solution to both equations. That method it is a solution to the system. The two lines space not identical, so the is the just solution.

D) (2, 7) is no a systems of one of two people equation, so the is no a equipment to the system

Incorrect. Substituting 2 because that x and also 7 because that y provides true declaration in both equations, therefore the point lies ~ above both lines. This method it is a equipment to both equations. It is likewise the just solution to the system.

You have the right to solve a device graphically. However, the is essential to remember the you must inspect the solution, together it could not it is in accurate.


Example

Problem

Find all solutions to the system y – x = 1 and y + x = 3.

*

First, graph both equations top top the exact same axes.

The 2 lines intersect once. That method there is only one equipment to the system.

The allude of intersection appears to be (1, 2).

Read the allude from the graph as accurately together possible.

y – x = 1

2 – 1 = 1

1 = 1

TRUE

(1, 2) is a solution of y – x = 1.

y + x = 3

2 + 1 = 3

3 = 3

TRUE

(1, 2) is a equipment of y + x = 3.

Check the values in both equations. Substitute 1 for x and also 2 because that y. (1, 2) is a solution.

Answer

(1, 2) is the systems to the device y – x = 1 and

y + x = 3.

Since (1, 2) is a solution for each of the equations in the system, that is the solution for the system.


Example

Problem

How numerous solutions go the device y = 2x + 1

and −4x + 2y = 2 have?

*

First, graph both equations top top the same axes.

The 2 equations graph as the exact same line. So every suggest on the line is a solution for the mechanism of equations.

Answer

The system y = 2x + 1 and −4x + 2y = 2 has an infinite variety of solutions.


Which point is the systems to the mechanism x – y = −1 and also 2x – y = −4? The device is graphed correctly below.

*

A) (−1, 2)

B) (−4, −3)

C) (−3, −2)

D) (−1, 1)


A) (−1, 2)

Incorrect. Substituting (−1, 2) into each equation, you uncover that that is a solution for 2x – y = −4, however not for x – y = −1. This way it cannot be a systems for the system. The correct answer is (−3, −2).

B) (−4, −3)

Incorrect. Substituting (−4, −3) right into each equation, you find that that is a systems for x – y = −1, however not because that 2x – y = −4. This method it can not be a equipment for the system. The correct answer is (−3, −2).

C) (−3, −2)

Correct. Substituting (−3, −2) into each equation shows this point is a equipment for both equations, so it is the systems for the system.

D) (−1, 1)

Incorrect. Substituting (−1, −1) into each equation, you find that the is neither a solution for 2x – y = −4, nor for x – y = −1. This method it cannot be a systems for the system. The correct answer is (−3, −2).

Graphing a system of equations for a real-world context can be beneficial in visualizing the problem. Stop look in ~ a couple of examples.


Example

Problem

In yesterday’s basketball game, Cheryl score 17 points through a mix of 2-point and also 3-point baskets. The variety of 2-point shots she made was one greater than the variety of 3-point shots she made. How countless of each type of basket did she score?

x = the variety of 2-point shots made

y = the variety of 3-point shots made

Assign variables to the 2 unknowns – the number of each kind of shots.

2x = the points from 2-point baskets

3y = the points indigenous 3-point baskets

Calculate how countless points room made from each of the two species of shots.

The variety of points Cheryl scored (17) =

the points from 2-point baskets + the points indigenous 3-point baskets.

17 = 2x + 3y

Write an equation utilizing information provided in the problem.

The number of 2-point baskets (x) = 1 + the number of 3-point baskets (y)

x = 1 + y

Write a second equation using extr information provided in the problem.

17 = 2x + 3y

x  = 1 + y

Now you have actually a device of two equations through two variables.

*

Graph both equations ~ above the very same axes.

The 2 lines intersect, so they have only one allude in common. That means there is only one solution to the system.

The allude of intersection appears to be (4, 3).

Read the allude of intersection from the graph.

17 = 2x+ 3y

17 = 2(4) + 3(3)

17 = 8 + 9

17 = 17

TRUE

(4, 3) is a solution of

17 = 2x + 3y.

x = 1 + y

4 = 1 + 3

4 = 4

TRUE

(4, 3) is a solution of

x = 1 + y

Check (4, 3) in each equation to see if it is a equipment to the mechanism of equations.

(4, 3) is a equipment to the equation.

x = 4 and also y = 3

Answer

Cheryl make 4 two-point baskets and also 3 three-point baskets.


Example

Problem

Andres to be trying to decide which of 2 mobile phone plan he should buy. One plan, TalkALot, fee a level fee that $15 per month for limitless minutes. One more plan, FriendFone, charged a monthly fee of $5 in enhancement to charging 20¢ every minute because that calls.

To examine the difference in plans, the made a graph:

*

If the plans to speak on the phone call for around 70 minutes per month, which plan should that purchase?

Look in ~ the graph. TalkALot is stood for as y = 15, if FriendFone is represented as

y = 0.2x + 5.

The number of minutes is listed on the x-axis. As soon as x = 70, TalkALot expenses $15, while FriendFone costs about $19.

Answer

Andres must buy theTalkALot plan.

Since TalkALot expenses less at 70 minutes, Andres must buy that plan.


Note that if the estimate had been incorrect, a brand-new estimate could have been made. Regraphing come zoom in top top the area whereby the lines cross would aid make a much better estimate.

Paco and also Lisel spent $30 going come the movies last night. Paco invested $8 much more than Lisel.

If ns = the amount that Paco spent, and L = the amount the Lisel spent, which mechanism of equations deserve to you use to figure out how much every of them spent?

A)

P + l = 30

P + 8 = L

B)

P + together = 30

P = l + 8

C)

P + 30 = L

P − 8 = L

D)

L + 30 = P

L − 8 = P


A)

P + l = 30

P + 8 = L

Incorrect. P + 8 = together reads: “Lisel invested $8 much more than Paco.” The correct mechanism is:

P + l = 30

P = l + 8

B)

P + together = 30

P = l + 8

Correct. The complete amount invested (P + L) is 30, therefore one equation must be ns + together = 30. Paco invested 8 dollars an ext than Lisel, so l + 8 will offer you the amount the Paco spent. This deserve to be rewritten p = together + 8.

See more: Which Term Identifies A Group Of Cells That Work Together, Introduction To Tissues

C)

P + 30 = L

P − 8 = L

Incorrect. P + 30 = together reads: “Lisel invested $30 much more than Paco.” The correct device is:

P + l = 30

P = l + 8

D)

L + 30 = P

L − 8 = P

Incorrect. Together + 30 = p reads: “Paco invested $30 much more than Lisel.” The correct system is:

P + l = 30

P = together + 8

A device of straight equations is 2 or much more linear equations that have the same variables. You can graph the equations as a mechanism to discover out even if it is the system has actually no services (represented by parallel lines), one solution (represented by intersecting lines), or an infinite number of solutions (represented by two superimposed lines). If graphing equipment of equations is a beneficial technique, relying top top graphs to recognize a certain point of intersection is not always an accurate method to uncover a specific solution for a system of equations.