Solve The System Of Equations Using The Elimination Method:${ \begin{array}{r} y = 4x + 10 \ -y = -5x - 8 \ \hline \end{array} }$

Introduction

The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in the equations are integers or can be easily manipulated to be integers. In this article, we will use the elimination method to solve a system of two linear equations in two variables.

The System of Equations

The system of equations we will be solving is:

$$\begin{array}{r} y = 4x + 10 \\ -y = -5x - 8 \\ \hline \end{array}$$

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same. In this case, we want to eliminate the variable $y$. We can multiply the first equation by $1$ and the second equation by $-1$.

$$\begin{array}{r} y = 4x + 10 \\ -(-y) = -(-5x - 8) \\ \hline \end{array}$$

Step 2: Add the Equations

Now that we have the coefficients of the variable to be eliminated as the same, we can add the equations to eliminate the variable $y$.

$$\begin{array}{r} y = 4x + 10 \\ y = 5x + 8 \\ \hline \end{array}$$

Step 3: Simplify the Equation

After adding the equations, we get a new equation with only one variable. We can simplify this equation by combining like terms.

$$\begin{array}{r} 4x + 10 = 5x + 8 \\ \hline \end{array}$$

Step 4: Solve for the Variable

Now that we have a simplified equation with only one variable, we can solve for the variable $x$.

$$\begin{array}{r} 4x + 10 = 5x + 8 \\ -10 = x - 8 \\ -18 = x \\ \hline \end{array}$$

Step 5: Find the Value of the Other Variable

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

$$\begin{array}{r} y = 4x + 10 \\ y = 4(-18) + 10 \\ y = -72 + 10 \\ y = -62 \\ \hline \end{array}$$

Conclusion

In this article, we used the elimination method to solve a system of two linear equations in two variables. We multiplied the equations by necessary multiples, added the equations, simplified the equation, solved for the variable, and found the value of the other variable. The solution to the system of equations is $x = -18$ and $y = -62$.

Example Use Cases

The elimination method can be used to solve systems of linear equations in various fields such as:

• Physics: To solve problems involving motion, forces, and energies.
• Engineering: To design and optimize systems, such as electrical circuits and mechanical systems.
• Computer Science: To solve problems involving algorithms and data structures.

Tips and Tricks

Here are some tips and tricks to keep in mind when using the elimination method:

• Choose the correct equation: Choose the equation that has the variable you want to eliminate.
• Multiply by necessary multiples: Multiply the equations by necessary multiples to eliminate the variable.
• Add the equations: Add the equations to eliminate the variable.
• Simplify the equation: Simplify the equation by combining like terms.
• Solve for the variable: Solve for the variable by isolating it on one side of the equation.

Conclusion

Q: What is the elimination method?

A: The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables.

Q: When should I use the elimination method?

A: You should use the elimination method when the coefficients of the variables in the equations are integers or can be easily manipulated to be integers.

Q: How do I choose the correct equation to eliminate?

A: To choose the correct equation to eliminate, look for the equation that has the variable you want to eliminate. If the coefficients of the variable are the same, you can eliminate the variable by adding or subtracting the equations.

Q: What if the coefficients of the variable are not the same?

A: If the coefficients of the variable are not the same, you can multiply the equations by necessary multiples to make the coefficients the same.

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

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

Q: What if I get a contradiction when using the elimination method?

A: If you get a contradiction when using the elimination method, it means that the system of equations has no solution.

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

A: Yes, you can use the elimination method to solve systems of equations with fractions or decimals. However, you may need to multiply the equations by necessary multiples to eliminate the fractions or decimals.

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

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

• Not choosing the correct equation to eliminate
• Not multiplying the equations by necessary multiples
• Not adding or subtracting the equations correctly
• Not simplifying the equation correctly

Q: How can I practice using the elimination method?

A: You can practice using the elimination method by solving systems of equations with different coefficients and variables. You can also use online resources or worksheets to practice.

Q: What are some real-world applications of the elimination method?

A: The elimination method has many real-world applications, including:

• Physics: To solve problems involving motion, forces, and energies.
• Engineering: To design and optimize systems, such as electrical circuits and mechanical systems.
• Computer Science: To solve problems involving algorithms and data structures.

Q: Can I use the elimination method to solve systems of equations with non-linear equations?

A: No, the elimination method is only used to solve systems of linear equations. If you have a system of equations with non-linear equations, you may need to use a different method, such as substitution or graphing.

Conclusion

The elimination method is a powerful technique used to solve systems of linear equations. By following the steps outlined in this article and practicing with different examples, you can become proficient in using the elimination method to solve systems of linear equations. Remember to choose the correct equation, multiply by necessary multiples, add the equations, simplify the equation, and solve for the variable. With practice and patience, you can become proficient in using the elimination method to solve systems of linear equations.