Solve The Following System Of Equations:${ \begin{aligned} x + 4y &= 17 \ -x - 6y &= -23 \end{aligned} }${ X =\$} { \square$}${ Y =\$} { \square$}$

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 method of substitution and elimination to find the values of the variables.

The System of Equations

The system of equations we will be solving is:

$$\begin{aligned} x + 4y &= 17 \\ -x - 6y &= -23 \end{aligned}$$

Step 1: Multiply the Two Equations

To solve this system of equations, we can multiply the two equations by necessary multiples such that the coefficients of $y$'s in both equations are the same.

Let's multiply the first equation by 3 and the second equation by 2:

$$\begin{aligned} 3(x + 4y) &= 3(17) \\ 2(-x - 6y) &= 2(-23) \end{aligned}$$

This gives us:

$$\begin{aligned} 3x + 12y &= 51 \\ -2x - 12y &= -46 \end{aligned}$$

Step 2: Add the Two Equations

Now, we can add the two equations to eliminate the variable $y$.

$$\begin{aligned} (3x + 12y) + (-2x - 12y) &= 51 + (-46) \\ 3x - 2x &= 5 \\ x &= 5 \end{aligned}$$

Step 3: Substitute the Value of $x$

Now that we have the value of $x$, we can substitute it into one of the original equations to find the value of $y$.

Let's substitute $x = 5$ into the first equation:

$$\begin{aligned} x + 4y &= 17 \\ 5 + 4y &= 17 \\ 4y &= 12 \\ y &= 3 \end{aligned}$$

Conclusion

In this article, we solved a system of two linear equations with two variables using the method of substitution and elimination. We multiplied the two equations by necessary multiples to eliminate the variable $y$, added the two equations to find the value of $x$, and then substituted the value of $x$ into one of the original equations to find the value of $y$.

Final Answer

The final answer is:

• $x = 5$
• $y = 3$

Discussion

This system of equations can be solved using other methods such as the method of substitution or the method of elimination. However, the method of elimination is often the most efficient method for solving systems of linear equations.

Example Use Cases

This method of solving systems of linear equations can be applied to a wide range of real-world problems, such as:

• Finding the intersection point of two lines
• Determining the amount of money in a bank account after a series of transactions
• Calculating the cost of a product after a series of discounts

Tips and Tricks

When solving systems of linear equations, it's often helpful to:

• Use the method of elimination to eliminate one of the variables
• Substitute the value of one variable into one of the original equations to find the value of the other variable
• Check your work by plugging the values of the variables back into the original equations

Conclusion

Introduction

In our previous article, we solved a system of two linear equations with two variables using the method of substitution and elimination. 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: 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: How do I know which method to use to solve a system of linear equations?

A: The method of substitution and elimination is often the most efficient method for solving systems of linear equations. However, the method of substitution may be more suitable for systems with two variables and two equations, while the method of elimination may be more suitable for systems with more variables and equations.

Q: What is the difference between the method of substitution and the method of elimination?

A: The method of substitution involves substituting the value of one variable into one of the original equations to find the value of the other variable. The method of elimination involves adding or subtracting the two equations to eliminate one of the variables.

Q: How do I know which variable to eliminate first?

A: To eliminate a variable, you need to have the same coefficient for that variable in both equations. You can multiply one or both equations by a constant to achieve this.

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

A: In this case, you can use the method of substitution and elimination to solve the system of equations. However, you may need to use more complex methods such as the method of matrices or the method of determinants.

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

A: Yes, you can use a calculator to solve a system 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?

A: 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, such as 2x + 3y = 5 and 2x + 3y = 7.

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

A: 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, such as 2x + 3y = 5 and 2x + 3y = 5.

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

A: Yes, you can use a graphing calculator to solve a system of linear equations. You can graph the two equations on the same coordinate plane and find the point of intersection.

Conclusion

In conclusion, solving a system of linear equations is a fundamental concept in mathematics that can be applied to a wide range of real-world problems. By using the method of substitution and elimination, we can find the values of the variables and solve the system of equations. We hope that this Q&A article has helped to clarify any questions you may have had about solving systems of linear equations.

Tips and Tricks

When solving systems of linear equations, it's often helpful to:

• Use the method of elimination to eliminate one of the variables
• Substitute the value of one variable into one of the original equations to find the value of the other variable
• Check your work by plugging the values of the variables back into the original equations
• Use a calculator or graphing calculator to check your work

Example Use Cases

This method of solving systems of linear equations can be applied to a wide range of real-world problems, such as:

• Finding the intersection point of two lines
• Determining the amount of money in a bank account after a series of transactions
• Calculating the cost of a product after a series of discounts

Discussion

This system of equations can be solved using other methods such as the method of substitution or the method of elimination. However, the method of elimination is often the most efficient method for solving systems of linear equations.

Final Answer

The final answer is:

• $x = 5$
• $y = 3$

References

• [1] "Solving Systems of Linear Equations" by Math Open Reference
• [2] "Solving Systems of Linear Equations" by Khan Academy
• [3] "Solving Systems of Linear Equations" by Wolfram MathWorld