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Introduction

Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving systems of equations using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation to solve for the remaining variable.

What is the Substitution Method?

The substitution method is a technique used to solve systems of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation to solve for the remaining variable. This method is particularly useful when one of the equations is easily solvable for one variable.

Step-by-Step Guide to Solving Systems of Equations by Substitution

To solve a system of equations using the substitution method, follow these steps:

  1. Choose one of the equations: Select one of the equations and solve it for one variable. This variable can be either x or y.
  2. Solve the equation for the chosen variable: Use algebraic manipulation to isolate the chosen variable on one side of the equation.
  3. Substitute the expression into the other equation: Take the expression you obtained in step 2 and substitute it into the other equation.
  4. Solve the resulting equation: Simplify the equation and solve for the remaining variable.
  5. Back-substitute to find the other variable: Once you have found the value of one variable, substitute it back into one of the original equations to find the value of the other variable.

Example: Solving a System of Equations by Substitution

Let's consider the following system of equations:

{ \begin{cases} 4x - 3y = 17 \\ 5x + 3y = 1 \end{cases} \}

To solve this system using the substitution method, we can follow the steps outlined above.

Step 1: Choose one of the equations

Let's choose the first equation and solve it for x:

4x−3y=174x - 3y = 17

We can add 3y to both sides of the equation to get:

4x=17+3y4x = 17 + 3y

Next, we can divide both sides of the equation by 4 to solve for x:

x=17+3y4x = \frac{17 + 3y}{4}

Step 2: Substitute the expression into the other equation

Now that we have an expression for x, we can substitute it into the second equation:

5x+3y=15x + 3y = 1

Substituting the expression for x, we get:

5(17+3y4)+3y=15\left(\frac{17 + 3y}{4}\right) + 3y = 1

Step 3: Solve the resulting equation

To simplify the equation, we can multiply both sides by 4 to get rid of the fraction:

5(17+3y)+12y=45(17 + 3y) + 12y = 4

Expanding the equation, we get:

85+15y+12y=485 + 15y + 12y = 4

Combine like terms:

85+27y=485 + 27y = 4

Subtract 85 from both sides:

27y=−8127y = -81

Divide both sides by 27:

y=−3y = -3

Step 4: Back-substitute to find the other variable

Now that we have found the value of y, we can substitute it back into one of the original equations to find the value of x. Let's use the first equation:

4x−3y=174x - 3y = 17

Substituting y = -3, we get:

4x−3(−3)=174x - 3(-3) = 17

Simplifying the equation, we get:

4x+9=174x + 9 = 17

Subtract 9 from both sides:

4x=84x = 8

Divide both sides by 4:

x=2x = 2

Conclusion

In this article, we have learned how to solve systems of equations using the substitution method. We have walked through a step-by-step guide to solving a system of equations and have applied the method to a specific example. By following these steps, you can solve systems of equations using the substitution method and become proficient in this important mathematical skill.

Tips and Tricks

  • When choosing an equation to solve for one variable, try to choose the one that is easiest to solve.
  • When substituting an expression into another equation, make sure to simplify the equation before solving for the remaining variable.
  • When back-substituting to find the other variable, make sure to use the correct value of the variable you found in the previous step.

Common Mistakes to Avoid

  • Failing to simplify the equation before solving for the remaining variable.
  • Failing to back-substitute to find the other variable.
  • Making algebraic errors when solving for the remaining variable.

Conclusion

Q: What is the substitution method?

A: The substitution method is a technique used to solve systems of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation to solve for the remaining variable.

Q: When should I use the substitution method?

A: You should use the substitution method when one of the equations is easily solvable for one variable. This method is particularly useful when you have a simple equation that can be solved quickly.

Q: How do I choose which equation to solve for one variable?

A: Choose the equation that is easiest to solve for one variable. If one equation has a simple coefficient or a straightforward solution, choose that one.

Q: What if I get stuck during the substitution process?

A: If you get stuck during the substitution process, try simplifying the equation before solving for the remaining variable. You can also try using a different equation or method to solve the system.

Q: Can I use the substitution method with non-linear equations?

A: No, the substitution method is typically used with linear equations. If you have a non-linear equation, you may need to use a different method, such as graphing or numerical methods.

Q: How do I know if I have found the correct solution?

A: To verify that you have found the correct solution, plug the values of x and y back into both original equations. If the equations are true, then you have found the correct solution.

Q: What if I get a system with no solution or infinitely many solutions?

A: If you get a system with no solution, it means that the equations are inconsistent and there is no solution. If you get a system with infinitely many solutions, it means that the equations are dependent and there are many possible solutions.

Q: Can I use the substitution method with systems of equations with more than two variables?

A: Yes, you can use the substitution method with systems of equations with more than two variables. However, it may be more complicated and require more steps.

Q: Are there any shortcuts or tricks for solving systems of equations by substitution?

A: Yes, there are several shortcuts and tricks that can make solving systems of equations by substitution easier. For example, you can use the method of elimination to eliminate one variable and then substitute the expression into the other equation.

Q: Can I use technology, such as calculators or computer software, to solve systems of equations by substitution?

A: Yes, you can use technology to solve systems of equations by substitution. Many calculators and computer software programs have built-in functions for solving systems of equations.

Conclusion

Solving systems of equations by substitution is a powerful tool for solving mathematical problems. By following the steps outlined in this article and using the FAQs to guide you, you can become proficient in this important mathematical skill. Remember to choose the easiest equation to solve for one variable, simplify the equation before solving for the remaining variable, and back-substitute to find the other variable. With practice and patience, you can master the substitution method and become a skilled mathematician.