Solve Using Substitution.$\[ \begin{array}{l} 4x - Y = -7 \\ -9x + Y = 17 \end{array} \\]

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Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. One of the methods used to solve a system of linear equations is 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 other variable. In this article, we will discuss how to solve a system of linear equations using the substitution method.

The Substitution Method

The substitution method is a simple and effective way to solve a system of linear equations. The steps involved in the substitution method are as follows:

  1. Solve one equation for one variable: Choose one of the equations and solve it for one of the variables. This will give you an expression for that variable in terms of the other variable.
  2. Substitute the expression into the other equation: Take the expression you obtained in step 1 and substitute it into the other equation. This will give you a new equation with only one variable.
  3. Solve for the variable: Solve the new equation for the variable. This will give you the value of the variable.
  4. Back-substitute to find the other variable: Once you have found the value of one variable, you can substitute it back into one of the original equations to find the value of the other variable.

Solving the Given System of Linear Equations

Let's use the substitution method to solve the following system of linear equations:

4x−y=−7−9x+y=17{ \begin{array}{l} 4x - y = -7 \\ -9x + y = 17 \end{array} }

Step 1: Solve one equation for one variable

We will solve the first equation for y:

4x−y=−7{ 4x - y = -7 }

y=4x+7{ y = 4x + 7 }

Step 2: Substitute the expression into the other equation

We will substitute the expression for y into the second equation:

−9x+y=17{ -9x + y = 17 }

−9x+(4x+7)=17{ -9x + (4x + 7) = 17 }

Step 3: Solve for the variable

We will simplify the equation and solve for x:

−9x+4x+7=17{ -9x + 4x + 7 = 17 }

−5x+7=17{ -5x + 7 = 17 }

−5x=10{ -5x = 10 }

x=−2{ x = -2 }

Step 4: Back-substitute to find the other variable

We will substitute the value of x back into one of the original equations to find the value of y:

4x−y=−7{ 4x - y = -7 }

4(−2)−y=−7{ 4(-2) - y = -7 }

−8−y=−7{ -8 - y = -7 }

−y=1{ -y = 1 }

y=−1{ y = -1 }

Conclusion

In this article, we discussed how to solve a system of linear equations using the substitution method. We used the substitution method to solve the following system of linear equations:

4x−y=−7−9x+y=17{ \begin{array}{l} 4x - y = -7 \\ -9x + y = 17 \end{array} }

We solved the system of linear equations by solving one equation for one variable, substituting the expression into the other equation, solving for the variable, and back-substituting to find the other variable. The solution to the system of linear equations is x = -2 and y = -1.

Example Problems

Here are a few example problems that you can try to practice the substitution method:

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

2x+3y=12x−2y=−3{ \begin{array}{l} 2x + 3y = 12 \\ x - 2y = -3 \end{array} }

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

x+2y=73x−2y=5{ \begin{array}{l} x + 2y = 7 \\ 3x - 2y = 5 \end{array} }

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

x−3y=−22x+3y=14{ \begin{array}{l} x - 3y = -2 \\ 2x + 3y = 14 \end{array} }

Tips and Tricks

Here are a few tips and tricks that you can use to help you solve systems of linear equations using the substitution method:

  1. Make sure to solve one equation for one variable before substituting the expression into the other equation.
  2. Make sure to simplify the equation before solving for the variable.
  3. Make sure to back-substitute to find the other variable once you have found the value of one variable.
  4. Make sure to check your solution by plugging it back into the original equations.

Conclusion

Q: What is the substitution method?

A: The substitution method is a technique used to solve systems of linear equations by substituting the expression for one variable into the other equation.

Q: How do I know which equation to solve for first?

A: You can choose either equation to solve for first. However, it's often easier to solve for the variable that appears in both equations.

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

A: If you get stuck, try simplifying the equation or checking your work. You can also try using a different method, such as the elimination method, to solve the system of linear equations.

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

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

Q: How do I know if the solution is correct?

A: To check if the solution is correct, plug the values of the variables back into the original equations and make sure they are true.

Q: What if the system of linear equations has no solution?

A: If the system of linear equations has no solution, it means that the equations are inconsistent and there is no value of the variables that can satisfy both equations.

Q: What if the system of linear equations has infinitely many solutions?

A: If the system of linear equations has infinitely many solutions, it means that the equations are dependent and there are many values of the variables that can satisfy both equations.

Q: Can I use the substitution method to solve systems of linear equations with fractions or decimals?

A: Yes, you can use the substitution method to solve systems of linear equations with fractions or decimals. However, you may need to simplify the equations and perform operations with fractions or decimals.

Q: How do I choose between the substitution method and the elimination method?

A: Both methods have their own advantages and disadvantages. The substitution method is often easier to use when one of the equations is already solved for one variable, while the elimination method is often easier to use when the coefficients of the variables are the same.

Q: Can I use the substitution method to solve systems of linear equations with variables on both sides of the equation?

A: Yes, you can use the substitution method to solve systems of linear equations with variables on both sides of the equation. However, you may need to simplify the equations and perform operations to isolate the variables.

Q: What are some common mistakes to avoid when using the substitution method?

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

  • Not solving one equation for one variable before substituting the expression into the other equation
  • Not simplifying the equation before solving for the variable
  • Not back-substituting to find the other variable once you have found the value of one variable
  • Not checking the solution by plugging it back into the original equations

Conclusion

In conclusion, the substitution method is a powerful technique for solving systems of linear equations. By following the steps involved in the substitution method and avoiding common mistakes, you can solve systems of linear equations and find the values of the variables. Remember to solve one equation for one variable, substitute the expression into the other equation, solve for the variable, and back-substitute to find the other variable. With practice, you will become proficient in using the substitution method to solve systems of linear equations.