Solve This System Of Equations:$\[ \begin{array}{c} 3x + 2y = 26 \\ y = 2x - 8 \end{array} \\]Write The Solution As A Coordinate Pair.

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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. In this article, we will focus on solving a system of two linear equations with two variables. We will use the substitution method to find the solution to the system of equations.

The System of Equations

The system of equations we will be solving is given by:

3x+2y=26y=2x−8\begin{array}{c} 3x + 2y = 26 \\ y = 2x - 8 \end{array}

Step 1: Write Down the Equations

The first equation is 3x+2y=263x + 2y = 26, and the second equation is y=2x−8y = 2x - 8. We will use the second equation to substitute the value of yy in the first equation.

Step 2: Substitute the Value of yy

We will substitute the value of yy from the second equation into the first equation. This gives us:

3x+2(2x−8)=263x + 2(2x - 8) = 26

Step 3: Simplify the Equation

We will simplify the equation by distributing the 2 to the terms inside the parentheses:

3x+4x−16=263x + 4x - 16 = 26

Step 4: Combine Like Terms

We will combine the like terms on the left-hand side of the equation:

7x−16=267x - 16 = 26

Step 5: Add 16 to Both Sides

We will add 16 to both sides of the equation to isolate the term with the variable:

7x=427x = 42

Step 6: Divide Both Sides by 7

We will divide both sides of the equation by 7 to solve for xx:

x=6x = 6

Step 7: Find the Value of yy

Now that we have the value of xx, we can substitute it into one of the original equations to find the value of yy. We will use the second equation:

y=2x−8y = 2x - 8

Substituting x=6x = 6 into the equation, we get:

y=2(6)−8y = 2(6) - 8

y=12−8y = 12 - 8

y=4y = 4

Step 8: Write the Solution as a Coordinate Pair

The solution to the system of equations is the coordinate pair (x,y)=(6,4)(x, y) = (6, 4).

Conclusion

In this article, we solved a system of two linear equations with two variables using the substitution method. We found the solution to the system of equations to be the coordinate pair (x,y)=(6,4)(x, y) = (6, 4). This method can be used to solve systems of linear equations with two or more variables.

Example Use Case

Solving systems of linear equations is a common problem in mathematics and has many real-world applications. For example, in economics, systems of linear equations can be used to model the behavior of supply and demand in a market. In engineering, systems of linear equations can be used to design and optimize systems such as electrical circuits and mechanical systems.

Tips and Tricks

When solving systems of linear equations, it is often helpful to use the substitution method or the elimination method. The substitution method involves substituting the value of one variable into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one of the variables. It is also helpful to check the solution by plugging it back into the original equations.

Further Reading

For more information on solving systems of linear equations, see the following resources:

References

  • [Boyer, C. B. (1968). A History of Mathematics. New York: Wiley.]
  • [Krantz, S. G. (1997). A First Course in Mathematical Modeling. Cambridge University Press.]
  • [Larson, R. E. (2009). Elementary and Intermediate Algebra: A Combined Approach. Cengage Learning.]

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Introduction

In our previous article, we solved a system of two linear equations with two variables using the substitution method. In this article, we will answer some frequently asked questions about solving systems of linear equations.

Q: What is a system of linear equations?

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.

Q: What are the different methods for solving systems of linear equations?

There are two main methods for solving systems of linear equations: the substitution method and the elimination method. The substitution method involves substituting the value of one variable into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one of the variables.

Q: What is the substitution method?

The substitution method involves substituting the value of one variable into the other equation. This is done by solving one of the equations for one of the variables and then substituting that value into the other equation.

Q: What is the elimination method?

The elimination method involves adding or subtracting the equations to eliminate one of the variables. This is done by multiplying one or both of the equations by a constant and then adding or subtracting the equations.

Q: How do I know which method to use?

The choice of method depends on the specific system of equations. If the equations are easy to solve, the substitution method may be the best choice. If the equations are more difficult to solve, the elimination method may be the best choice.

Q: What if I have a system of three or more linear equations?

If you have a system of three or more linear equations, you can use the same methods as before, but you may need to use a combination of the substitution and elimination methods.

Q: Can I use a calculator to solve systems of linear equations?

Yes, you can use a calculator to solve systems of linear equations. Many calculators have built-in functions for solving systems of linear equations.

Q: What if I have a system of linear equations with no solution?

If you have a system of linear equations with no solution, it means that the equations are inconsistent. This can happen if the equations are contradictory or if the system is inconsistent.

Q: What if I have a system of linear equations with infinitely many solutions?

If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent. This can happen if the equations are identical or if the system is dependent.

Q: How do I check my solution?

To check your solution, plug the values back into the original equations and make sure they are true.

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

Solving systems of linear equations has many real-world applications, including:

  • Modeling the behavior of supply and demand in a market
  • Designing and optimizing systems such as electrical circuits and mechanical systems
  • Solving problems in physics and engineering
  • Analyzing data in statistics and economics

Conclusion

Solving systems of linear equations is an important skill in mathematics and has many real-world applications. By understanding the different methods for solving systems of linear equations, you can solve a wide range of problems and make informed decisions.

Example Use Case

Solving systems of linear equations is a common problem in mathematics and has many real-world applications. For example, in economics, systems of linear equations can be used to model the behavior of supply and demand in a market. In engineering, systems of linear equations can be used to design and optimize systems such as electrical circuits and mechanical systems.

Tips and Tricks

When solving systems of linear equations, it is often helpful to use the substitution method or the elimination method. The substitution method involves substituting the value of one variable into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one of the variables. It is also helpful to check the solution by plugging it back into the original equations.

Further Reading

For more information on solving systems of linear equations, see the following resources:

References

  • [Boyer, C. B. (1968). A History of Mathematics. New York: Wiley.]
  • [Krantz, S. G. (1997). A First Course in Mathematical Modeling. Cambridge University Press.]
  • [Larson, R. E. (2009). Elementary and Intermediate Algebra: A Combined Approach. Cengage Learning.]