· determine whether a mechanism of linear equations is continuous or inconsistent.

You are watching: Which system of equations is consistent and dependent?

· determine whether a mechanism of straight equations is dependency or independent.

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

· fix application difficulties by graphing a mechanism of equations.


Recall the a direct equation graphs as a line, which indicates that every one of the points on the heat are solutions to that direct equation. There space an infinite number of solutions. If you have actually a system of linear equations, the systems for the mechanism 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 works with of this point will be the equipment for the 2 variables in the two equations.


The solution for a mechanism of equations is the value or worths that are true for all equations in the system. The graphs that equations in ~ a system deserve to tell friend how numerous solutions exist for that system. Look in ~ the photos below. Each shows two lines that consist of a mechanism of equations.

One Solution

No Solutions

Infinite Solutions

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*

*

If the graphs of the equations intersect, then there is one solution that is true because that both equations.

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

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

When the currently intersect, the point of intersection is the only point that the 2 graphs have actually in common. Therefore the collaborates of that suggest are the systems for the 2 variables used in the equations. Once the lines room parallel, there space no solutions, and sometimes the 2 equations will graph together the very same line, in which situation we have actually an infinite variety of solutions.

Some unique terms are sometimes used to explain these kinds of systems.

The complying with terms refer to how countless solutions the device has.

o as soon as a system has actually one equipment (the graphs that the equations crossing once), the device is a consistent mechanism of linear equations and the equations space independent.

o once a system has actually no equipment (the graphs that the equations don’t crossing at all), the device is one inconsistent system of direct equations and also the equations space independent.

o If the lines space the very same (the graphs intersect at every points), the system is a continual system of straight equations and also the equations space dependent. That is, any type of solution that one equation must also be a systems of the other, so the equations depend on each other.

The adhering to terms describe whether the device has any solutions in ~ all.

o The device is a continuous system of straight equations when it has solutions.

o The system is an inconsistent mechanism of straight equations as soon as it has actually no solutions.

We can summarize this together follows:

o A system with one or an ext solutions is consistent.

o A mechanism with no solutions is inconsistent.

o If the lines are different, the equations room independent straight equations.

o If the lines room the same, the equations space dependent linear equations.


Example

Problem

Using the graph the y = x and x + 2y = 6, presented below, determine how countless solutions the mechanism has. Climate classify the mechanism as consistent or inconsistent and also the equations together dependent or independent.

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The lines intersect at one point. For this reason the 2 lines have actually only one allude in common, over there is just one solution to the system.

Because the lines space not the very same the equations are independent.

Because there is just one solution, this mechanism is consistent.

Answer

The mechanism is consistent and the equations room independent.


Advanced Example

Problem

Using the graph of y = 3.5x + 0.25 and also 14x – 4y = -4.5, displayed below, determine how many solutions the device has. Then classify the system as continual or inconsistent and the equations as dependent or independent.

*

The lines space parallel, an interpretation they perform not intersect. There space no services to the system.

The lines space not the same, the equations space independent.

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

Answer

The system is inconsistent and also the equations room independent.


Advanced Question

Which of the complying with represents dependent equations and consistent systems?

A)

B)

C)

D)


A)

Incorrect. The 2 lines in this system have actually the exact same slope, yet different values for b. This way the lines are parallel. The lines don’t intersect, so there space no solutions and also the system is inconsistent. Due to the fact that the lines space not the exact same the equations space independent. The exactly answer is C.

B)

Incorrect. The 2 lines in this device have various slopes and different values for b. This way the lines intersect at one point. Because there is a solution, this system is consistent. And also because the lines are not the same, the equations are independent. The exactly answer is C.

C)

Correct. The two lines in this mechanism are the same;  can it is in rewritten as . Because there are many solutions, this system is consistent. The present are the same so the equations room dependent.

D)

Incorrect. The two lines in this device have different slopes and also the exact same value for b. This method the lines crossing at one point—the y-intercept. Recall that intersecting lines have actually one solution and therefore the mechanism is consistent. Because the lines room not the very same the equations room independent. The exactly answer is C.

From the graph above, you can see that there is one systems to the mechanism y = x and also x + 2y = 6. The solution shows up to be (2, 2). However, you must verify an answer that you read from a graph come be certain that it’s no really (2.001, 2.001) or (1.9943, 1.9943).

One method of verifying the the suggest does exist ~ above both currently is to instead of the x- and also y-values of the ordered pair right into the equation of every line. If the substitution outcomes in a true statement, climate you have actually the exactly solution!


Example

Problem

Is (2, 2) a equipment of the device y = x and x + 2y = 6?

y = x

2 = 2

TRUE

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

x + 2y = 6

2 + 2(2) = 6

2 + 4 = 6

6 = 6

TRUE

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

Since the systems of the mechanism must be a equipment to every the equations in the system, check the suggest in every equation. Substitute 2 because that x and also 2 for y in every equation.

Answer

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

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


Example

Problem

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

y = 3x

9 = 3(3)

TRUE

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

2x – y = 6

2(3) – 9 = 6

6 – 9 = 6

-3 = 6

FALSE

(3, 9) is no a systems of 2x – y = 6.

Since the systems of the mechanism must be a systems to every the equations in the system, inspect the point in every equation. Substitute 3 because that x and 9 because that y in each equation.

Answer

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

Since (3, 9) is no a solution of among the equations in the system, it can not be a systems of the system.


Example

Problem

Is (−2, 4) a solution of the device y = 2x and also 3x + 2y = 1?

y = 2x

4 = 2(−2)

4 = −4

FALSE

(−2, 4) is not a solution of y = 2x.

3x + 2y = 1

3(−2) + 2(4) = 1

−6 + 8 = 1

2 = 1

FALSE

(−2, 4) is no a systems of 3x + 2y = 1.

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

Answer

(−2, 4) is no a equipment to the system.

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


Remember, that in stimulate to it is in a systems to the mechanism of equations, the value of the point must it is in a solution for both equations. As soon as you discover one equation because that which the point is false, girlfriend have figured out that that is no a systems for the system.

Which of the following statements is true because that the device 2x – y = −3 and y = 4x – 1?

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

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

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

D) (2, 7) is no a equipment of one of two people equation, so that is not a equipment to the system


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

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

B) (2, 7) is a equipment of one equation however not the other, so it is not a systems of the system

Incorrect. The suggest (2, 7) is a equipment of both equations, so the is a systems of the system. The two lines room not identical, so the is the just solution.

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

Correct. Substituting 2 because that x and also 7 because that y provides true declaration in both equations, for this reason the allude is a solution to both equations. That method it is a equipment to the system. The 2 lines room not identical, so the is the just solution.

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

Incorrect. Substituting 2 because that x and also 7 for y offers true explanation in both equations, so the suggest lies ~ above both lines. This way it is a solution to both equations. It is additionally the just solution to the system.

You deserve to solve a device graphically. However, it is crucial to remember that you must check the solution, as it could not be accurate.


Example

Problem

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

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First, graph both equations on the very same axes.

The two lines crossing once. That means there is just one equipment to the system.

The point of intersection appears to it is in (1, 2).

Read the point from the graph as accurately as 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 systems of y + x = 3.

Check the worths in both equations. Substitute 1 for x and 2 for y. (1, 2) is a solution.

Answer

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

y + x = 3.

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


Example

Problem

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

and −4x + 2y = 2 have?

*

First, graph both equations ~ above the very same axes.

The 2 equations graph together the exact same line. For this reason every suggest on that line is a equipment for the system of equations.

Answer

The mechanism y = 2x + 1 and also −4x + 2y = 2 has actually an infinite number of solutions.


Which point is the solution to the system x – y = −1 and also 2x – y = −4? The device is graphed properly below.

*

A) (−1, 2)

B) (−4, −3)

C) (−3, −2)

D) (−1, 1)


A) (−1, 2)

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

B) (−4, −3)

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

C) (−3, −2)

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

D) (−1, 1)

Incorrect. Substituting (−1, −1) into each equation, you find that that is no a solution for 2x – y = −4, nor because that x – y = −1. This way it can not be a systems for the system. The exactly answer is (−3, −2).

Graphing a system of equations for a real-world context can be beneficial in visualizing the problem. Let’s look at a couple of examples.


Example

Problem

In yesterday’s basketball game, Cheryl score 17 points with a combination of 2-point and 3-point baskets. The variety of 2-point shots she made to be one greater than the number of 3-point shots she made. How plenty of of each kind of basket did she score?

x = the number of 2-point shots made

y = the variety of 3-point shots made

Assign variables come the 2 unknowns – the variety of each form of shots.

2x = the points from 2-point baskets

3y = the points indigenous 3-point baskets

Calculate how plenty of points space made from each of the two varieties of shots.

The variety of points Cheryl scored (17) =

the points native 2-point baskets + the points native 3-point baskets.

17 = 2x + 3y

Write one equation making use of information offered in the problem.

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

x = 1 + y

Write a 2nd equation using added information offered in the problem.

17 = 2x + 3y

x  = 1 + y

Now you have actually a mechanism of 2 equations v two variables.

*

Graph both equations top top the same axes.

The 2 lines intersect, therefore they have actually only one allude in common. That way there is only one systems to the system.

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

Read the suggest of intersection indigenous the graph.

17 = 2x+ 3y

17 = 2(4) + 3(3)

17 = 8 + 9

17 = 17

TRUE

(4, 3) is a systems of

17 = 2x + 3y.

x = 1 + y

4 = 1 + 3

4 = 4

TRUE

(4, 3) is a equipment of

x = 1 + y

Check (4, 3) in each equation to see if the is a solution to the device of equations.

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

x = 4 and also y = 3

Answer

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


Example

Problem

Andres to be trying to decision which of two mobile phone plan he need to buy. One plan, TalkALot, fee a level fee of $15 per month for limitless minutes. An additional plan, FriendFone, charged a monthly fee of $5 in addition to charging 20¢ per minute because that calls.

To research the distinction in plans, the made a graph:

*

If he plans to talk on the phone call for about 70 minutes per month, which arrangement should he purchase?

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

y = 0.2x + 5.

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

Answer

Andres need to buy theTalkALot plan.

Since TalkALot costs less in ~ 70 minutes, Andres should buy the plan.


Note the if the estimate had been incorrect, a brand-new estimate might have been made. Regraphing come zoom in on the area whereby the lines cross would help make a better estimate.

Paco and Lisel invested $30 going come the movies critical night. Paco spent $8 an ext than Lisel.

If ns = the amount that Paco spent, and also L = the amount the Lisel spent, which system of equations can you usage to number out just how much every of lock spent?

A)

P + together = 30

P + 8 = L

B)

P + l = 30

P = l + 8

C)

P + 30 = L

P − 8 = L

D)

L + 30 = P

L − 8 = P


A)

P + together = 30

P + 8 = L

Incorrect. P + 8 = together reads: “Lisel spent $8 an ext than Paco.” The correct mechanism is:

P + together = 30

P = together + 8

B)

P + together = 30

P = l + 8

Correct. The full amount invested (P + L) is 30, therefore one equation need to be ns + l = 30. Paco invested 8 dollars much more than Lisel, so l + 8 will offer you the amount the Paco spent. This have the right to be rewritten ns = together + 8.

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C)

P + 30 = L

P − 8 = L

Incorrect. Ns + 30 = l reads: “Lisel invested $30 an ext than Paco.” The correct system is:

P + together = 30

P = together + 8

D)

L + 30 = P

L − 8 = P

Incorrect. Together + 30 = p reads: “Paco spent $30 an ext than Lisel.” The correct system is:

P + l = 30

P = l + 8

A mechanism of direct equations is 2 or an ext linear equations that have actually the same variables. You have the right to graph the equations as a mechanism to discover out whether the system has actually no options (represented by parallel lines), one equipment (represented through intersecting lines), or one infinite variety of solutions (represented by two superimposed lines). While graphing solution of equations is a beneficial technique, relying ~ above graphs to identify a particular point of intersection is not constantly an accurate way to find a precise solution for a mechanism of equations.