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Linear Systems

Dinner is booked for half an hour after the movie. When then, is dinner booked for? It's impossible to know, you don't have enough information! What if, however, you are told that the movie ends at six o'clock? Well, then you probably know intuitively that dinner must be at half-past six. You have just instinctively solved a linear system, and probably didn't even realise!

A linear system is a group of linear equations involving the same variables. There is no limit to the number of equations a linear system can have.

These linear systems are an important aspect of mathematics, that can be used to describe real-world scenarios as well as more abstract problems such as in linear algebra. In this article, you'll learn how to use linear equations to build linear systems, and how to solve such systems.

Let's consider the equations andtogether they form a linear system. By plotting each equation on a graph, we can get a visual overview of the linear system as a whole.

Example of a linear system, John Hannah - StudySmarter Originals

How to Build a Linear System

Often, we might be presented with a problem or real-world scenario that is actually a linear system. If we can recognize that a linear system is being described, and have the correct information at our disposal, then we can build it by expressing it algebraically. Let's take a look at an example to see how this is done.

A woman buys concert tickets for her three children, as well as an adult ticket for herself. The total cost of her tickets is $£100.$ Her friend buys a ticket each for himself and his spouse to the same concert, as well as two tickets for their children. Her friend paid $£120$ in total for his tickets.

Solution:

Firstly, how can we recognize that this is a linear system? Well, the observant may notice that there are two variables in the scenario, common to both purchasers: the cost of a child's ticket and the cost of an adult's ticket. After all, a linear system is just a group of linear equations involving the same variables.

Now, how do we actually build our linear system from this information? We start by labelling each variable we have discerned. Let's say that x is the cost of a child's ticket, and $y$ is the cost of an adult's ticket. From here we simply construct two equations from the information above.

The information we are given says three child's tickets and one adult's cost $£100$ total, therefore...



Similarly, we are told that two child's tickets and 2 adult's tickets cost $£120 in$total, therefore...



And with those equations, we have just built our first linear system!



How to Solve Linear Systems

Linear systems are useful because they can be solved. For instance, these solutions can be used to tell when one runner in a race might overtake the other, or how much was paid each for an apple and banana at the shop.

A solution to a linear system is the assignment of values to each variable such that all equations in the system hold true. When visualized on a graph, this is the point where the lines of all of the equations cross.

Let's consider the linear system we just built concerning the adults' and children's concert tickets.

The solution to this system, presented as an ordered pair, is $\left(20,40\right). This$communicates that a child's ticket costs $£20,$ and an adult's costs $£40.$ Try plugging these values in for $x$ and $y$ to prove that the equations hold true.

Types of Linear Systems

Any linear system can be categorized as one of two types depending on the number of solutions it possesses; it is said to be consistent or inconsistent.

A linear system is said to be consistent if it has one or more solutions. Furthermore, a dependent linear system has infinite solutions, and an independent system has a unique solution.

The following linear system is said to be consistent and independent as it has a unique solution, which is (1, 3). We can tell that this system has a solution as we can clearly see that there is one point where all three lines intersect.

Graph of a consistent, independent linear system, John Hannah - StudySmarter Originals

The following linear system is said to be consistent and dependent, as it has an infinite number of solutions. When graphed it appears as a single line, however, there are two equations in this system and they are equal at all points.

Graph of a consistent, dependent linear system, John Hannah - StudySmarter

A linear system is said to be inconsistent if it has no solutions.

The following linear system is said to be inconsistent as it has no solution. We can tell that this system has no solution as we can clearly see that there isn't a point where all three lines intersect. This is because there are no values ofand for which all three equations hold true.

Graph of an inconsistent linear system, John Hannah - StudySmarter Originals

Linear Systems Examples

Which of the following are linear systems?

(a)

(b)

(c)

(d)

Solutions:

(a) Not linear system (b) Linear system (c) Linear system (d) Linear system

Classify the following linear systems as dependent consistent, independent consistent, or inconsistent.

(a)

Types of linear systems example, John Hannah - StudySmarter Originals

(b)

Types of linear systems example, John Hannah - StudySmarter Originals

(c)

Types of linear systems example, John Hannah - StudySmarter Originals

Solutions:

(a) Inconsistent (b) Consistent, Independent (c) Consistent, dependent

Linear Systems - Key takeaways

• Linear systems are collections of linear equations that share the same variables.
• There is no limit to the number of equations or variables these linear systems can contain.
• Given enough information, it is possible to build a linear system from a real-world scenario.
• A solution to a linear system is the assignment of values to each variable such that all equations in the system hold true.
• A linear system has can one solution, an infinite number of solutions, or no solution.

Linear systems are collections of linear equations with the same variables.

An example of a linear system is:

y = 2x + 1

y = 3x + 2

Linear systems can be solved using simultaneous equations, matrices, linear combinations, or graphs.

A solution to a linear system is the assignment of values to each variable such that all equations in the system hold true.

Linear systems can be categorised as consistent or inconsistent. A linear system is consistent if it has at least one solution and inconsistent if it has no solutions.

Final Linear Systems Quiz

Question

What are vectors?

Vectors are quantities that possess both magnitude and direction.

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Question

Quantities are those that possess only magnitudes are known as___

Scalar quantities

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Question

Vectors possess position. Is this statement true or false?

False

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Question

All the options are examples of vector quantities except

Speed

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Question

A vector that has a value magnitude of one is known as ___

A unit vector

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Question

What mathematical functions are used to find the values of the components of vectors?

Trigonometric functions

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Question

With the two-dimensional coordinate system, vectors can be decomposed into what components?

the horizontal (x-component) and the vertical (y-component)

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Question

Calculate (9, -1, 6) + (2, 8, -3)

(11, 7, 3)

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Question

Solving linear systems by substitution involves how many variables?

Two (x and y)

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Question

The objective of solving linear systems by the substitution method is to make one _____ the subject of one of the equations

Variable

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Question

What is done next after one variable is made the subject of the equation?

It is substituted into the other equation

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Question

What is the first step in dealing with linear systems involving fractions?

Eliminate the fractions first

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Question

Find the solution to 4 - 3x < 10

x > -2

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Question

A dashed line is used to graph any two-variable inequality that contains either of these symbols.

When a two-variable inequality contains either less than (<) or greater than (>), a dashed line is used to graph the boundary line.

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Question

Find the solution to x + 4 < 12

The solution is x < 8

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Question

Find the solution to 2n - 9 > 1

The solution is n > 5

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Question

Represent the following in interval notation. x > -3 and x < 5

The interval notation is (-3, 5)

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Question

Which ordered pair makes both inequalities true?

y > -3x + 3

y > 2x - 2

(2.2)

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Question

Which ordered pair makes both inequalities true?

y > -2x + 3

y < x - 2

(3.0)

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Question

A linear __________ is a linear expression with two variables that uses <, >, <, >

Inequality

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Question

Which ordered pair makes both inequalities true?

y > -3x + 3

y > 2x - 2

(2,2)

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Question

What condition must be met before adding or subtracting matrices?

Matrices must be of the same dimensions.

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Question

What condition must be met before matrices are multiplied?

The number of columns of the first matrix must be equal to the number of rows of the second matrix.

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Question

What is an identity matrix?

An identity matrix is a square matrix in which when multiplied by another square matrix equals to the same matrix. In this matrix, the elements from the topmost left diagonal to the downmost right diagonal is 1 while every other element in the matrix is 0.

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Question

What is the inverse of a matrix?

A matrix is said to be the inverse of another matrix if the product of both matrices result in an identity matrix.

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Question

What methods are used in finding the inverse of a matrix?

The Gaussian method and the determinant method

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Question

What two conditions must be met for a matrix to have an inverse?

The matrix must be a square matrix; also, the determinant of the matrix must not be 0.

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Question

How many equations would you require at a minimum to solve a system of simultaneous equations involving 5 unknown variables?

5

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Question

How many equations would you require at a minimum to solve a system of simultaneous equations involving 2 unknown variables?

2

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Question

Find the values of a and b:

5a - 3b = -9

2a + b = 14

a = 3, b = 8

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Question

Find the values of x and y:

4x - 9y = 10

2x - 3y = 8

x = 7, y = 2

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Question

Find the values of x and y:

3x + 3y = 12

x + 2y = 9

x = -1, y = 5

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Question

Find the values of a and b:

-2a + 4b = 16

8a + 3b = 12

a = 0, b = 4

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Question

State whether the following statement is true or false:

It is possible for a linear system in 3 variables to have 0 solutions.

True

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Question

State whether the following statement is true or false:

It is possible for a linear system in 2 variables to have 4 solutions.

False

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Question

State whether the following statement is true or false:

It is possible for a linear system in 2 variables to have infinite solutions.

true

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Question

State whether the following statement is true or false:

It is possible for a linear system in 3 variables to have infinite solutions.

True

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Question

State whether the following statement is true or false:

It is possible for a linear system in 3 variables to have 1 unique solution.

True

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Question

Find the values of x and y:

3x + y = 4

x - y = 4

x = 2, y = -2

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Question

Find the values of x and y:

4x + y = 3

2x + 5y = -3

x = 1, y = -1

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Question

List some methods used in solving linear systems.

• Substitution method
• Elimination method
• Graphing method
• Graphs and tables

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Question

Solve the following linear system using the substitution method, \left\{\begin{align}x+y&=2\\4y+2&=x\end{align}\right.

$$x=2$$ and $$y=0$$

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Question

What is another name for the elimination method?

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Question

It is possible to get the solution of a linear system using a table of values alone?

True

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Question

When solving linear systems by linear algebra, you must convert the linear equations to matrices.

True

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Question

Which of these is NOT part of the steps used in solving linear systems by linear algebra?

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Question

Which of the following is required when using the elimination method?

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Question

When making the table of values, use only positive numbers.

False

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Question

What happens when the coefficients are opposite in the elimination method?

We add the equations to eliminate one variable.

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Question

After getting the value of one variable in the substitution method, what do you do next?

You substitute in another equation to get the value of the other variable present.

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