Solving Systems of Equations by Substitution Worksheet, Examples, and Video

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How to Solve Systems by Substitution in 4 EASY steps (Substitution Method Examples)

Key Points about Solving Systems by Substitution

  • Substitution method is used to solve systems of equations by solving one equation for one variable and then substituting the expression into the other equation(s) and solving for the other variable(s).
  • A system of equations is a set of two or more equations with two or more variables. The solution to a system of equations is the set of values that make all the equations in the system true.
  • Solving systems of equations by substitution requires a good understanding of systems of equations and the ability to solve equations for one variable.

Solving Systems of Equations by Substitution: A Complete Guide

Solving Systems of Equations by Substitution is a method to solve a system of two linear equations. Solving Systems of Equations by Substitution follows a specific process in order to simplify the solutions. The first thing you must do when Solving Systems of Equations by Substitution is to solve one equation for either variable. Next, you must take the solution for the variable and substitute it into the other equation for the variable. Then you solve for the variable and get a numerical solution. Use the solution of the variable to substitute back in to either original equation. The last step when finding out how to solve Systems of Equations by Substitution is to solve for the last variable by following order of operations.

Solving systems of equations by substitution is a fundamental concept in algebra. It is used to find the values of two or more variables that satisfy a set of equations. Substitution method involves solving one equation for one variable and then substituting the expression into the other equation(s) and solving for the other variable(s). This method is useful when one of the equations is already solved for one variable.

To solve a system of equations by substitution, one must have a good understanding of systems of equations. A system of equations is a set of two or more equations with two or more variables. The solution to a system of equations is the set of values that make all the equations in the system true. A system of equations can have one solution, no solution, or infinitely many solutions.

In this article, we will explore how to solve systems of equations by substitution. We will provide step-by-step instructions and examples to help readers understand the concept. We will also discuss word problems that involve substitution method and answer frequently asked questions about the topic.

Common Core Standard: 8.EE.C
Related Topics: Identifying One, None, Infinite Solutions, Solving Systems by Graphing, Solving Systems by Elimination
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Solving Systems of Equations by Substitution

How to Solve Systems of Equations by Substitution

The substitution method is a powerful tool for solving systems of equations. It involves solving one equation for one variable, and then substituting that expression into the other equation. This results in a single equation with just one variable, which can be solved using basic algebraic techniques. Here’s how to solve systems of equations by substitution:

  1. Identify the two equations that make up the system. These equations will typically be in the form of y = mx + b, where m is the slope of the line and b is the y-intercept.
  2. Choose one of the equations and solve it for one of the variables. It’s usually best to choose the equation that has the simpler variable to solve for. For example, if one equation is y = 2x + 3 and the other equation is y = -x + 5, it would be easier to solve for x in the first equation.
  3. Substitute the expression you just solved for into the other equation. This will result in a single equation with just one variable.
  4. Solve the resulting equation using basic algebraic techniques. This will give you the value of the variable you just solved for.
  5. Substitute the value you just found back into one of the original equations to find the value of the other variable.
  6. Check your solution by substituting both values back into both original equations. If the values satisfy both equations, then you have found the correct solution.

The substitution method is a powerful tool for solving systems of equations, but it can be time-consuming and tedious. It’s important to be patient and careful when using this method, and to check your work carefully to ensure that you have found the correct solution. With practice, however, you’ll find that solving systems of equations by substitution becomes easier and more intuitive.

 

Understanding Systems of Equations

When solving problems involving multiple variables, a system of equations is often used. A system of equations is a set of two or more equations that must be solved simultaneously. Each equation in the system represents a relationship between variables.

A system of equations can be linear or nonlinear. A linear system of equations is a set of equations where each equation is a linear equation. A linear equation is an equation that can be written in the form of y = mx + b, where m and b are constants, and x and y are variables.

Systems of Equations Substitution Method

One way to solve a system of linear equations is by using the substitution method. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This will result in an equation with only one variable, which can then be solved.

Here are the steps to solve a system of linear equations using the substitution method:

  1. Choose one equation in the system and solve for one of the variables in terms of the other variable.
  2. Substitute the expression obtained in step 1 into the other equation in the system.
  3. Solve the resulting equation for the remaining variable.
  4. Substitute the value obtained in step 3 into either of the original equations to find the value of the other variable.
  5. Write the solution as an ordered pair.

Practice

To get a better understanding of the substitution method, it is important to practice solving systems of equations with this method. Here are some practice problems:

  1. Solve the system of equations using the substitution method:

    x + y = 5
    x – y = 1

  2. Solve the system of equations using the substitution method:

    2x + y = 7
    x – y = 1

  3. Solve the system of equations using the substitution method:

    3x – 2y = 8
    x + y = 5

By practicing these problems, you can get a better understanding of how to use the substitution method to solve systems of equations.

 

Solving Systems of Equations by Substitution Solution

2 Simple Solving Systems of Equations by Substitution Examples

When solving a system of equations by substitution, the goal is to find the value of each variable that satisfies both equations. Here are a few examples of how to solve systems of equations by substitution.

Example 1

Consider the system of equations:

2x + y = 5
x - y = 1

To solve this system by substitution, follow these steps:

  1. Solve one of the equations for one of the variables. In this case, we can solve the second equation for x:
x = y + 1
  1. Substitute the expression for x into the other equation. In this case, we substitute y + 1 for x in the first equation:
2(y + 1) + y = 5
  1. Simplify the resulting equation by combining like terms:
3y + 2 = 5
  1. Solve for y:
y = 1
  1. Substitute the value of y into one of the original equations to find x. We can use the second equation:
x - 1 = 1
x = 2
  1. Check the solution by substituting the values of x and y into both equations. The solution is (2, 1).

Checking the Solution

To check the solution, substitute x = 2 and y = 1 into both equations:

2(2) + 1 = 5
2 - 1 = 1

Both equations are true, so the solution is correct.

Example 2

Consider the system of equations:

3x - y = 2
x + 2y = 8

To solve this system by substitution, follow these steps:

  1. Solve one of the equations for one of the variables. In this case, we can solve the first equation for y:
y = 3x - 2
  1. Substitute the expression for y into the other equation. In this case, we substitute 3x - 2 for y in the second equation:
x + 2(3x - 2) = 8
  1. Simplify the resulting equation by combining like terms:
7x - 4 = 8
  1. Solve for x:
x = 2
  1. Substitute the value of x into one of the original equations to find y. We can use the first equation:
3(2) - y = 2
y = 4
  1. Check the solution by substituting the values of x and y into both equations. The solution is (2, 4).

Checking the Solution

To check the solution, substitute x = 2 and y = 4 into both equations:

3(2) - 4 = 2
2 + 2(4) = 8

Both equations are true, so the solution is correct.

 

5 Quick Systems of Equations Substitution Method Practice Problems

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Solving Systems by Substitution Quiz

Click Start to begin the practice quiz!

1 / 5

Solve the system using the Substitution Method.

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2 / 5

Solve the system using the Substitution Method.

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3 / 5

Solve the system using the Substitution Method.

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4 / 5

Solve the system using the Substitution Method.

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5 / 5

Solve the system using the Substitution Method.

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Systems of Equation Substitution Word Problems

Systems of equation substitution is a powerful tool that can be used to solve a variety of real-world problems. In these types of problems, there are typically two or more unknown variables that need to be solved for. By setting up a system of equations and using substitution to solve for one variable in terms of another, it is possible to find the values of all the variables.

Applications of Systems of Equations

Systems of equations can be used to solve a wide range of problems in many different fields. Some common applications include:

  • Business: Systems of equations can be used to model supply and demand, profit and loss, and other financial problems.
  • Science: Systems of equations can be used to model physical phenomena, such as the motion of objects or the behavior of chemical reactions.
  • Engineering: Systems of equations can be used to design and optimize systems, such as electrical circuits or mechanical structures.
  • Social sciences: Systems of equations can be used to model human behavior, such as voting patterns or economic trends.

Problem Solving Strategy

When solving a system of equations using substitution, it is important to follow a clear strategy to ensure that the correct solution is obtained. The following steps can be used as a guide:

  1. Identify the two equations that need to be solved.
  2. Choose one of the equations and solve for one variable in terms of the other. This will give an expression that can be substituted into the other equation.
  3. Substitute the expression from step 2 into the other equation and solve for the remaining variable.
  4. Substitute the value found in step 3 back into either of the original equations to find the value of the other variable.
  5. Check the solution by plugging the values into both equations to make sure they are satisfied.

By following this strategy, it is possible to solve a wide range of problems using systems of equation substitution.

 

Solving Systems Using Substitution FAQ

What is the substitution method for solving systems of equations?

The substitution method is a technique used to solve a system of equations by expressing one variable in terms of the other variable in one of the equations and then substituting that expression into the other equation. This method allows us to eliminate one variable and solve for the other.

How do you solve a system of equations by substitution?

To solve a system of equations by substitution, follow these steps:

  1. Choose one of the equations and solve for one of the variables in terms of the other variable.
  2. Substitute the expression found in step 1 into the other equation.
  3. Solve the resulting equation for the remaining variable.
  4. Substitute the value found in step 3 into either of the original equations to find the value of the other variable.

What is the difference between substitution and elimination methods for solving systems of equations?

The substitution method involves solving one of the equations for one of the variables and then substituting that expression into the other equation to eliminate one variable and solve for the other. The elimination method involves adding or subtracting the equations to eliminate one variable and solve for the other. Both methods can be used to solve systems of equations, but the choice of method depends on the specific problem.

Can you solve any system of equations by substitution?

No, not all systems of equations can be solved by substitution. Some systems may have no solution or an infinite number of solutions.

How do you know when to use substitution or elimination method to solve a system of equations?

The choice of method depends on the specific problem. If one of the equations can be easily solved for one of the variables, the substitution method may be more efficient. If the coefficients of one of the variables are opposite in the two equations, the elimination method may be more efficient.

What are some common mistakes to avoid when using the substitution method to solve systems of equations?

Some common mistakes to avoid when using the substitution method include:

  • Forgetting to substitute the expression found in step 1 into the other equation.
  • Making arithmetic errors when solving the resulting equation for the remaining variable.
  • Forgetting to substitute the value found in step 3 into either of the original equations to find the value of the other variable.

How do you know that substitution gives the answer to a system of equations?

To check if the solution found using the substitution method is correct, substitute the values of the variables into both equations and verify that both equations are true. If both equations are true, the solution is correct.

 

Solving Systems of Equations by Substitution Worksheet: Video Explanation

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

Watch the free Solving Systems of Equations by Substitution video on YouTube here: How to Solve Systems of Equations by Substitution

Video Transcript:
This video is about solving systems of equations substitution. You can get the solving systems by substitution worksheet​ used in this video for free by clicking on our link in the description below.

Here we are at a first problem. Anytime you’re trying to solve a system of equations your answer must be in the form of an X and a y-coordinate. That means anytime you solve this by any method whether it’s graphing elimination or substitution you have to have a coordinate as your answer. When solving systems like this you will solve by substitution using the substitution method algebra. That means you already have to know how to solve by substitution.

Here we learn our first problem on our systems of equations substitution worksheet. We have 3x minus y equals 7 and then y equals 2x plus 2. Now when you’re solving systems by substitution you have to use substitution to substitute one equation in to the other equation. In the case of this first problem, our second equation here is already solved for the variable Y.

We already know Y is equal to 2x plus 2 because this is already solved for y. We can take 2x + 2 and we can substitute it in for Y in the other equation. What we’re going to do is we’re going to take our top equation and we’re going to rewrite it with our second equation substitute it into it. We have our second equation which is 2x + 2 and we know Y is equal to that.

We substituted it in for where Y used to be now we have to solve for X. That means we have to simplify this. We’re going to go ahead and distribute this negative to everything inside the parenthesis. Now we have 3x minus 2x minus 2 because we do negative 1 times 2x and then negative 1 times 2 equals 7. Now the next step is to combine like terms. We do 3x minus 2x and 3x minus 2x is 1/x. You bring down your -2 and your 7 and then we’re going to add 2 to both sides so that we cancel this too and we end up with 1x or just x equals 9.

Now we know that X is equal to 9 and you remember you have to have an X and a y-coordinate. We take our X is equal to 9 and we’re going to substitute that back in to X in our bottom equation. We’ll take y equals 2 times our x value which is 9 plus 2  y equals 2 times 9 which is 18 plus 2. Y equals 18 plus 2 which is 20. Now our complete solution is x equals 9. Our x coordinate is 9 and then our y coordinate is 20. We’ll use X is 9 y is 20 and that is the coordinate that is our solution.

The next problem we’re going to go over on our system on our solving systems of equations by substitution worksheet is number 3. Now this time we have negative x plus 3y equals 3 and then the second equation is X minus 2y equals 8. Now this time we don’t have any variable that has already been solved for. What we have to do is we have to pick a variable to solve for so that once it’s solved for we can substitute it into the other equation.

You can solve for any variable,  it’s up to you. You could solve for this one you could solve for this Y, you could solve for this X or you could solve for this 2y. Easiest thing to solve for would be to solve for this X down here because all you have to do is add 2y to both sides and these will cancel and you solve for x automatically.

Now we know X is equal to these two terms if we tried to solve for any other variables we would have had a coefficient here and we would have had to divide by everything by that coefficient.  This was just the easiest variable to solve for. We know x is equal to 2y plus 8 we have to take this 2y plus 8 and we have to substitute it to our second equation or top equation in this case.  When we do that we’re going to take our equation which is negative x or in this case negative 2y plus 8 because that’s what we’re substituting in for and then plus 3y.

We bring the rest over. Now we know it’s equal to negative 2y plus 8. We take this negative and we distribute it to everything on the inside of the parenthesis. This will be negative 2y negative 1 times 8 minus 8 plus 3y equals 3. The next step is to combine like terms so we’re going to add these two together negative 2y plus 3y is 1y. You bring down your minus 8 and then you bring down your equals 3. Then we have to add 8 to both sides you get Y. 1y or just y equals 11.

We know Y is equal to 11 so now we’re going to take y equals 11 and we’re going to substitute 11 in for the Y back in the equation we simplified for X. We have x equals 2 times we know Y is 11 so we’re going to substitute 11 in for y plus 8 2 times 11 is 22 Plus 8. And then when you solve 22 Plus 8 you will get 30. The solution because we have to have a coordinate as x equals 30 y is 11 and that’s our total solution. You can try these practice problems by downloading the free solving system of equations by substitution worksheet above.

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