Systems of Linear Equations Worksheet, Examples, and Word Problems

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Key Points about Systems of Linear Equations

Systems of Linear Equations involve multiple linear equations that work together to solve for the variables.
There are multiple methods to solve systems of linear equations, such as graphing, substitution, and elimination.
Understanding systems of linear equations is crucial for solving real-world problems in various fields of study.

Solving Systems of Linear Equations in Two Variables

There are three different types of solutions to Systems of Linear Equations. There can be One Solution, No Solution, and Infinite Solutions. When Identifying One, None, and Infinite Solutions to equations you must first solve the equation for the variable. When solving for the variable, you need to follow the rules and properties for solving equations. Once the equation has been solved, there can be the One Solution, No Solution, or Infinite Solutions.

Identifying One, None, and Infinite Solutions to Systems of Linear Equations can be simple once the equation has been solved for the variable. For the equation to have One Solution, your answer will be a variable equals a number. For the equation to have No Solution, your answer will be an Untrue Statement of numbers. For the equation to have Infinite Solutions, your answer will be a True Statement of numbers.

Systems of Linear Equations are a fundamental part of algebra and are used extensively in various fields of study, including physics, engineering, and economics. A system of linear equations is a collection of two or more linear equations that involve the same variables. The goal is to find the values of the variables that satisfy all the equations in the system simultaneously.

Solving systems of linear equations can be accomplished through various methods, such as graphing, substitution, and elimination. Graphing involves plotting the equations on a graph and finding the point where they intersect. Substitution involves solving one equation for one variable and plugging that expression into the other equations. Elimination involves adding or subtracting the equations to eliminate one of the variables. The method used depends on the specific problem and personal preference.

Understanding systems of linear equations is essential in solving real-world problems, such as determining the optimal production levels of goods or the most efficient route for a delivery truck. Being able to solve these problems requires a solid understanding of the underlying concepts and the ability to apply them in a practical setting.

Common Core Standard: 8.EE.C.8
Related Topics: Solving Systems by Graphing, Solving Systems by Elimination, Solving Systems by Substitution
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Systems of Linear Equations

What are Systems of Linear Equations?

A System of Linear Equations is a collection of two or more linear equations that are considered together. These equations are used to solve problems that involve multiple unknowns. The solutions to these systems of equations are the values of the variables that make all the equations true.

Basic Concepts

A linear equation is an equation where the variables are raised to the first power and there are no products of variables. A system of linear equations is when two or more linear equations are considered together. These equations can be solved using various methods, such as substitution or elimination.

Variables in Linear Equations

In linear equations, the variables are usually represented by letters such as x, y, and z. These variables can take on any real number value. The coefficients in front of the variables are also real numbers.

Definition and Types

A system of linear equations can be defined as a set of equations where each equation is a linear equation. There are different types of systems of linear equations, such as homogeneous and non-homogeneous systems. In a homogeneous system, all the constants on the right-hand side of the equations are zero. In a non-homogeneous system, at least one constant is non-zero.

Consistent and Inconsistent Systems

A system of linear equations can be classified as consistent or inconsistent. A consistent system has at least one solution that satisfies all the equations in the system. An inconsistent system has no solutions that satisfy all the equations in the system.

Dependent and Independent Systems

A system of linear equations can also be classified as dependent or independent. A dependent system has infinitely many solutions. An independent system has a unique solution.

In conclusion, Systems of Linear Equations are used to solve problems that involve multiple unknowns. These systems consist of two or more linear equations and can be solved using various methods. The solutions to these systems are the values of the variables that make all the equations true.

How to Solve Systems of Linear Equations

Solving systems of linear equations can be done using various methods such as substitution, elimination, or graphing. A system of linear equations is a set of two or more linear equations with the same variables. The goal is to find the values of the variables that satisfy all the equations in the system.

One way to solve a system of linear equations is by using substitution. This method involves solving one equation for one variable and then substituting that expression into the other equation. The resulting equation will have only one variable, which can be solved using basic algebraic operations. The solution pair obtained can then be substituted into the original equations to check if they hold true.

Another method is elimination, also known as addition. This method involves adding or subtracting the equations in the system to eliminate one of the variables. The resulting equation will have only one variable, which can be solved using basic algebraic operations. The solution pair obtained can then be substituted into the original equations to check if they hold true.

It is important to note that a system of linear equations can have one solution, no solution, or infinitely many solutions. A system with one solution means that there is only one set of values for the variables that satisfy all the equations in the system. A system with no solution means that there are no values for the variables that satisfy all the equations in the system. A system with infinitely many solutions means that there are many sets of values for the variables that satisfy all the equations in the system.

When solving a system of linear equations with two variables, the solution pair can be represented as an ordered pair (x,y). The type of system can be classified as either dependent or independent. An independent system has only one solution, while a dependent system has infinitely many solutions.

In summary, solving systems of linear equations involves finding the values of the variables that satisfy all the equations in the system. This can be done using methods such as substitution or elimination. It is important to check the solution pair obtained by substituting it into the original equations. A system can have one solution, no solution, or infinitely many solutions. The solution pair can be represented as an ordered pair (x,y), and the type of system can be classified as either dependent or independent.

Systems of Linear Equations Solution

3 Simple Systems of Linear Equations Examples

Systems of linear equations are used to solve problems that involve two or more linear equations.

Solve the System of Linear Equations for the variable.
If you get x equals a number as your answer, then there is one solution to the system.
If you get a number equals a different number, then there are no solutions to the system.
If you get a number equals the same number, then there are infinite solutions to the system.

Here are a few examples of how systems of linear equations can be used in real-world situations:

Example 1: Cost of Apples and Oranges

Suppose a person buys some apples and oranges from a grocery store. The cost of each apple is $0.50, and the cost of each orange is $0.75. The person buys a total of 10 fruits and spends a total of $5.25. How many apples and oranges did the person buy?

Let x be the number of apples and y be the number of oranges. Then the following system of equations can be used to solve the problem:

0.50x + 0.75y = 5.25
x + y = 10

Solving this system of equations gives x = 6 and y = 4. Therefore, the person bought 6 apples and 4 oranges.

Example 2: Speed of a Boat and a Current

Suppose a boat travels 10 miles upstream in 2 hours and then travels 10 miles downstream in 1 hour. If the speed of the current is constant, what is the speed of the boat and the speed of the current?

Let x be the speed of the boat and y be the speed of the current. Then the following system of equations can be used to solve the problem:

x - y = 5
x + y = 10

Solving this system of equations gives x = 7.5 and y = 2.5. Therefore, the speed of the boat is 7.5 mph and the speed of the current is 2.5 mph.

Example 3: Investment in Stocks and Bonds

Suppose an investor has $10,000 to invest in stocks and bonds. The expected rate of return on stocks is 8% per year, and the expected rate of return on bonds is 5% per year. The investor wants to earn a total of $700 per year from the investment. How much should the investor invest in stocks and how much should be invested in bonds?

Let x be the amount invested in stocks and y be the amount invested in bonds. Then the following system of equations can be used to solve the problem:

0.08x + 0.05y = 700
x + y = 10000

Solving this system of equations gives x = 4000 and y = 6000. Therefore, the investor should invest $4,000 in stocks and $6,000 in bonds to earn a total of $700 per year from the investment.

5 Quick Solving Systems of Equations Practice Problems

Graphing Systems of Linear Equations

Graphing systems of linear equations is a method to find the solution of two linear equations in two variables. This method involves plotting the two equations on the same graph and finding the point of intersection of the two lines. The point of intersection represents the solution of the system of equations.

Plotting Linear Systems

To plot a linear system, one needs to graph both equations on the same coordinate plane. The x-axis represents the horizontal axis, and the y-axis represents the vertical axis. Each equation can be represented as a line on the graph.

To plot a line, one needs to find two points on the line. One can find these points by selecting any two values for x and then solving for y. Once these two points are found, one can plot them on the graph and draw a straight line through them.

Interpretation of Graphs

The intersection of the two lines represents a solution that satisfies both equations. There are three possible outcomes when graphing a system of linear equations:

If the two lines intersect at one point, then there is one unique solution to the system of equations.
If the two lines are parallel, then there is no solution to the system of equations because they never intersect.
If the two lines coincide, then there are infinite solutions to the system of equations because every point on the line satisfies both equations.

One can also use the slope of the line to interpret the graph. The slope of a line represents the ratio of the change in y to the change in x. If the slope of two lines is the same, then they are parallel. If the slope of two lines is different, then they intersect at some point.

Graphing systems of linear equations is a useful method to find the solution of two linear equations in two variables. It is especially helpful when the equations are given in slope-intercept form. This method allows one to visualize the solution and interpret the graph to determine the number of solutions and the values of the variables.

Systems of Linear Equations Word Problems

Linear equations are a fundamental part of algebra and are widely used in various fields, including science, engineering, and economics. One of the most important applications of linear equations is in solving real-world problems. In this section, we will discuss systems of linear equations word problems and how to solve them.

Real World Examples

Linear equations can be used to model many real-world scenarios. For example, a skateboard manufacturer might want to know how much profit they will make on a particular model. To do this, they need to consider the revenue and costs associated with producing and selling the skateboard. Using linear equations, they can create a system of equations to solve for the unknown variables.

Another example could be a chemistry experiment where the researcher needs to calculate the concentration of a solution. They can use linear equations to relate the amount of solute to the volume of the solution and solve for the unknown concentration.

Problem Solving

To solve systems of linear equations word problems, one needs to follow a systematic approach. The first step is to define the variables and write down the equations that relate them. The equations can be solved using various techniques, such as substitution or elimination.

For example, consider the following problem: “A store sells two types of pens, blue and red. The total revenue from selling 30 blue pens and 20 red pens is $150. Also, the revenue from selling 10 blue pens and 30 red pens is $120. What is the cost of one blue pen and one red pen?”

To solve this problem, one can define the variables, let x be the cost of one blue pen and y be the cost of one red pen. Using this, we can write two equations:

30x + 20y = 150

10x + 30y = 120

These equations can be solved using elimination or substitution to find the values of x and y.

Linear equations can also be represented using matrices, which can make solving systems of equations more efficient. Matrices are used extensively in various fields, including physics, engineering, and computer science.

In conclusion, systems of linear equations word problems are an essential part of real-world problem-solving. By defining variables, writing equations, and using appropriate techniques, one can solve these problems and gain valuable insights.

Solving Systems of Linear Equations FAQ

What are some common methods for solving systems of linear equations?

There are several methods for solving systems of linear equations, including substitution, elimination, and graphing. Substitution involves solving one equation for one variable and then plugging that expression into the other equation. Elimination involves adding or subtracting the equations to eliminate one of the variables. Graphing involves graphing each equation and finding the point where the two lines intersect.

What are the different types of solutions for systems of linear equations?

There are three types of solutions for systems of linear equations: one unique solution, infinitely many solutions, and no solution. A system has one unique solution if the two lines intersect at one point. A system has infinitely many solutions if the two lines are the same, meaning they coincide. A system has no solution if the two lines are parallel and never intersect.

How can you tell if a system of linear equations has no solution?

A system of linear equations has no solution if the two lines are parallel and never intersect. You can determine this by looking at the slopes of the two lines. If the slopes are equal, the lines are parallel and there is no solution.

What is the difference between consistent and inconsistent systems of linear equations?

A consistent system of linear equations has at least one solution, while an inconsistent system has no solution. A system can be consistent and have one unique solution, or it can be consistent and have infinitely many solutions.

What is the relationship between matrices and systems of linear equations?

Matrices can be used to represent systems of linear equations. The coefficients of the variables are placed in a matrix, and the constant terms are placed in another matrix. The two matrices are then combined and manipulated using matrix operations to solve for the variables.

What are some real-world applications of systems of linear equations?

Systems of linear equations are used in many real-world applications, such as optimizing production in manufacturing, determining the best investment portfolio, and predicting the behavior of complex systems in physics and engineering.

What are 4 methods of solving linear systems?

The four methods of solving linear systems are substitution, elimination, graphing, and matrices. Substitution involves solving one equation for one variable and plugging it into the other equation. Elimination involves adding or subtracting the equations to eliminate one of the variables. Graphing involves graphing each equation and finding the point where the two lines intersect. Matrices involve representing the system of equations as a matrix and using matrix operations to solve for the variables.

What are the 3 solutions of systems of linear equation?

The three solutions of systems of linear equations are one unique solution, infinitely many solutions, and no solution. A system has one unique solution if the two lines intersect at one point. A system has infinitely many solutions if the two lines coincide. A system has no solution if the two lines are parallel and never intersect.

Systems of Linear Equations Worksheet Video Explanation

Watch our free video on how to solve Systems of Linear Equations. This video shows how to solve problems that are on our free Solving Systems of Linear Equations worksheet that you can get by submitting your email above.

Watch the free Systems of Linear Equations video on YouTube here: Systems of Linear Equations

Video Transcript:

This video is about systems of linear equations. You can get the one solution no solution infinite solutions worksheet we use in this video for free by clicking on the link in the description below.

Here we are the first problem for systems of linear equations. We need to state whether this system has one solution, no solutions, or an infinite amount of solutions. In order to determine what type of solution this system has, you have to know the difference between the three types. If it has one solution, that means you’re going to have an answer where x equals a number. It would look like any typical answer that you get for an equation for a system.

To have no solutions that would mean that you get an untrue statement. What this means is you would get some number, let’s say in this case 17, is equal to something that it is not, which would be let’s say 8. This is not true 17 does not equal 8. That means that the system would have no solutions. Then finally the last type would be an infinite amount of solutions, and that is when you get a true statement. This means that you could have a number that equals the same number. If we had 17 equals 17, that is true right, 17 doesn’t equal 17 so that means it would be an infinite amount of solutions.

Looking at our problem, we’re given 12 X plus 6 equals 2x minus 24. We have to solve this equation for X. When we go to solve this the first thing we need to do is we need to get all of the variables on one side and then we need to get all the constants on the other side. The first thing we’re going to do is we’re going to subtract 2x from both sides so that we get the X is only on one side.

12x minus 2x is 10x then you bring down your plus 6 and you bring down your minus 24. Then we have to get the constants on the opposite side of the variable. We will subtract 6 from both sides, these will cancel and you end up with and X is equal to negative 30. Then the last step is to divide both sides by 10. When we do this we will get x equals negative 3.

Our answer is x equals negative 3. That means this has one solution to the system of linear equations because we know that X is equal to negative 3 and only negative 3. Our answer can only be negative 3 which means it’s one solution. This is a system one solution example problem.

Moving on to the second problem for system of linear equations, we are given 3 x plus 15 equals 3x plus 15. Now the first step to solve this is we have to get the variables on one side together. What we’re going to do is we’re going to subtract this 3x and then we’re going to subtract 3x from this side. When we do this the 3 X’s here cancel and also the 3 X’s here cancel. We are left with 15 equals 15. We bring these down. Our X’s are cancelled and we’re left with just two constants.

In this case the constants are equal to each other. 15 equals 15. When you have two constants that are equal to each other that means that the answer for the system of linear equations is infinite solution example. That means no matter what you put in for X you will always get a true statement. You could put any number in for X and you will always get something that is true. Because it’s always true you can use any number and that means you have an infinite amount of answers that will be true or that will answer this system. This is a system of equations infinite solutions example problem.

The last problem we’re going to go over is number 4. Number 4 gives us the system 3x plus 18 equals 3 times the quantity X minus 6. The first step in this equation is to distribute the 3 to both the X and the negative 6. When we do this we will multiply 3 times X which is 3x and then 3 times negative 6 which is negative 18. The next step is to get all the constants on one side together. We’re going to subtract this 3x over here and we’re also going to subtract this 3x over here. When we do that these X’s cancel and these X’s also cancel. We’re going to bring down the 18 on this side and on this side we have negative 18. Our solution is 18 equals negative 18.

That is not true that is an untrue statement 18 does not equal negative 18. They are different numbers that means that this system has no solution. There’s no number that you can substitute in for X that will get you a true statement. That means there are no solutions to this. This is a system of equations no solutions example problem.Try all the practice problems by downloading the free one solution, no solution, infinite solutions worksheet above.