Use The Method Of Substitution To Solve The Following System Of Equations. If The System Is Dependent, Express The Solution Set In Terms Of One Of The Variables. Leave All Fractional Answers In Fraction Form.$\[ \begin{cases} x + 6y = -23 \\ 4x +

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Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. The method of substitution is one of the techniques used to solve systems of equations. 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 use the method of substitution to solve a system of equations and express the solution set in terms of one of the variables.

The Method of Substitution

The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation to solve for the other variable. To use this method, we need to follow these steps:

  1. Solve one equation for one variable.
  2. Substitute the expression from step 1 into the other equation.
  3. Solve the resulting equation for the other variable.
  4. Express the solution set in terms of one of the variables.

Solving the System of Equations

Let's consider the following system of equations:

{x+6y=−234x+2y=−46{ \begin{cases} x + 6y = -23 \\ 4x + 2y = -46 \end{cases} }

To solve this system of equations using the method of substitution, we will first solve the first equation for x.

Solving the First Equation for x

We can solve the first equation for x by subtracting 6y from both sides of the equation and then dividing both sides by 1.

x+6y=−23{ x + 6y = -23 }

x=−23−6y{ x = -23 - 6y }

Now that we have solved the first equation for x, we can substitute this expression into the second equation.

Substituting the Expression into the Second Equation

We will substitute the expression x=−23−6yx = -23 - 6y into the second equation.

4x+2y=−46{ 4x + 2y = -46 }

4(−23−6y)+2y=−46{ 4(-23 - 6y) + 2y = -46 }

−92−24y+2y=−46{ -92 - 24y + 2y = -46 }

−92−22y=−46{ -92 - 22y = -46 }

−22y=46−92{ -22y = 46 - 92 }

−22y=−46{ -22y = -46 }

y=−46−22{ y = \frac{-46}{-22} }

y=2311{ y = \frac{23}{11} }

Now that we have found the value of y, we can substitute this value back into the expression x=−23−6yx = -23 - 6y to find the value of x.

Finding the Value of x

We will substitute the value of y back into the expression x=−23−6yx = -23 - 6y.

x=−23−6y{ x = -23 - 6y }

x=−23−6(2311){ x = -23 - 6\left(\frac{23}{11}\right) }

x=−23−13811{ x = -23 - \frac{138}{11} }

x=−23−13811{ x = -23 - \frac{138}{11} }

x=−253−13811{ x = \frac{-253 - 138}{11} }

x=−39111{ x = \frac{-391}{11} }

x=−39111{ x = -\frac{391}{11} }

Expressing the Solution Set in Terms of One of the Variables

The solution set is x=−39111x = -\frac{391}{11} and y=2311y = \frac{23}{11}. We can express the solution set in terms of one of the variables by substituting the value of x into one of the original equations.

Let's substitute the value of x into the first equation.

x+6y=−23{ x + 6y = -23 }

−39111+6y=−23{ -\frac{391}{11} + 6y = -23 }

6y=−23+39111{ 6y = -23 + \frac{391}{11} }

6y=−253+39111{ 6y = \frac{-253 + 391}{11} }

6y=13811{ 6y = \frac{138}{11} }

y=13811÷6{ y = \frac{138}{11} \div 6 }

y=13866{ y = \frac{138}{66} }

y=2311{ y = \frac{23}{11} }

As we can see, the solution set is x=−39111x = -\frac{391}{11} and y=2311y = \frac{23}{11}.

Conclusion

Q: What is the method of substitution?

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

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 a fraction as an answer?

A: If you get a fraction as an answer, you can leave it in fraction form or simplify it to a decimal or mixed number.

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

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

Q: What if I get a system of equations that has no solution?

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

Q: Can I use the method of substitution with systems of equations that have dependent variables?

A: Yes, you can use the method of substitution with systems of equations that have dependent variables. In this case, the solution will be an equation that relates the variables.

Q: How do I know if a system of equations has dependent variables?

A: A system of equations has dependent variables if the equations are equivalent and can be written as multiples of each other.

Q: What if I get a system of equations that has infinitely many solutions?

A: If you get a system of equations that has infinitely many solutions, it means that the equations are equivalent and can be written as multiples of each other.

Q: Can I use the method of substitution with systems of equations that have infinitely many solutions?

A: Yes, you can use the method of substitution with systems of equations that have infinitely many solutions. In this case, the solution will be an equation that relates the variables.

Q: How do I know if a system of equations has infinitely many solutions?

A: A system of equations has infinitely many solutions if the equations are equivalent and can be written as multiples of each other.

Q: What if I get a system of equations that has a single solution?

A: If you get a system of equations that has a single solution, it means that the equations are consistent and there is a unique value of the variables that can satisfy both equations.

Q: Can I use the method of substitution with systems of equations that have a single solution?

A: Yes, you can use the method of substitution with systems of equations that have a single solution. In this case, the solution will be a specific value of the variables.

Conclusion

In this article, we answered some frequently asked questions about solving systems of equations using the method of substitution. We covered topics such as choosing which equation to solve for first, dealing with fractions, and determining if a system of equations has dependent variables, infinitely many solutions, or a single solution. We hope that this article has been helpful in answering your questions and providing a better understanding of the method of substitution.