Adding the Modified First Equation to the Second Equation

Oct 13, 2025 by miodeya.com 1 views

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. When dealing with a system of linear equations, it is often necessary to eliminate one of the variables to solve for the other. In this article, we will examine a system of two linear equations and determine an equivalent form of the first equation that, when added to the second equation, eliminates the y terms.

The System of Equations

The given system of equations is:

${-\frac{5}{8}x + \frac{3}{4}y = 12}$
${8x + 12y = 24}$

Understanding the Goal

Our goal is to find an equivalent form of the first equation that, when added to the second equation, eliminates the y terms. This means that we want to manipulate the first equation in such a way that the coefficient of y becomes zero when we add it to the second equation.

Method 1: Multiply the First Equation by a Suitable Constant

One way to eliminate the y terms is to multiply the first equation by a suitable constant. Let's try multiplying the first equation by 4, which is the least common multiple of the denominators of the coefficients of y.

Multiplying the first equation by 4 gives us:

${-\frac{5}{2}x + 3y = 48}$

Adding the Modified First Equation to the Second Equation

Now, let's add the modified first equation to the second equation:

${-\frac{5}{2}x + 3y + 8x + 12y = 48 + 24}$

Simplifying the equation, we get:

${\frac{11}{2}x + 15y = 72}$

However, we still have a y term in the equation. To eliminate the y term, we need to multiply the entire equation by a suitable constant.

Method 2: Multiply the Entire Equation by a Suitable Constant

Let's multiply the entire equation by -2/15, which is the reciprocal of the coefficient of y.

Multiplying the equation by -2/15 gives us:

${-\frac{11}{2}x - 2y = -\frac{2}{5} \times 72}$

Simplifying the equation, we get:

${-\frac{11}{2}x - 2y = -\frac{144}{5}}$

Adding the Modified Equation to the Second Equation

Now, let's add the modified equation to the second equation:

${-\frac{5}{8}x + \frac{3}{4}y + 8x + 12y = 12 - \frac{144}{5}}$

Simplifying the equation, we get:

${\frac{61}{8}x = -\frac{108}{5}}$

However, we still have an x term in the equation. To eliminate the x term, we need to multiply the entire equation by a suitable constant.

Method 3: Multiply the Entire Equation by a Suitable Constant

Let's multiply the entire equation by 8/61, which is the reciprocal of the coefficient of x.

Multiplying the equation by 8/61 gives us:

${x = -\frac{8}{5} \times \frac{108}{61}}$

Simplifying the equation, we get:

${x = -\frac{864}{305}}$

Conclusion

In this article, we examined a system of two linear equations and determined an equivalent form of the first equation that, when added to the second equation, eliminates the y terms. We used three different methods to achieve this goal, including multiplying the first equation by a suitable constant, multiplying the entire equation by a suitable constant, and adding the modified equation to the second equation. By following these methods, we were able to eliminate the y terms and solve for the value of x.

Final Answer

The final answer is:

${x = -\frac{864}{305}}$

Note

The value of y can be found by substituting the value of x into one of the original equations. However, since our goal was to eliminate the y terms, we did not pursue this further.

References

[1] "Systems of Linear Equations" by Paul's Online Math Notes
[2] "Linear Algebra and Its Applications" by Gilbert Strang

Additional Resources

Khan Academy: Systems of Linear Equations
MIT OpenCourseWare: Linear Algebra
Wolfram Alpha: Systems of Linear Equations
Examine the System of Equations: Q&A =====================================

Introduction

In our previous article, we examined a system of two linear equations and determined an equivalent form of the first equation that, when added to the second equation, eliminates the y terms. In this article, we will answer some frequently asked questions related to the system of equations and provide additional insights.

Q: What is the purpose of eliminating the y terms?

A: The purpose of eliminating the y terms is to solve for the value of x. By eliminating the y terms, we can isolate the x term and solve for its value.

Q: How do I know which method to use to eliminate the y terms?

A: There are several methods to eliminate the y terms, including multiplying the first equation by a suitable constant, multiplying the entire equation by a suitable constant, and adding the modified equation to the second equation. The choice of method depends on the specific equation and the desired outcome.

Q: Can I use any constant to eliminate the y terms?

A: No, you cannot use any constant to eliminate the y terms. The constant must be chosen such that it eliminates the y term when added to the second equation.

Q: How do I choose the constant to eliminate the y terms?

A: To choose the constant, you need to find the least common multiple of the denominators of the coefficients of y. This will ensure that the y term is eliminated when the modified equation is added to the second equation.

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

A: If you have a system of three or more equations, you can use the same methods to eliminate the y terms. However, you may need to use multiple steps to eliminate all the y terms.

Q: Can I use substitution or elimination to solve the system of equations?

A: Yes, you can use substitution or elimination to solve the system of equations. Substitution involves solving one equation for one variable and then substituting that value into the other equation. Elimination involves adding or subtracting the equations to eliminate one variable.

Q: What is the difference between substitution and elimination?

A: The main difference between substitution and elimination is the method used to eliminate the y terms. Substitution involves solving one equation for one variable and then substituting that value into the other equation. Elimination involves adding or subtracting the equations to eliminate one variable.

Q: Can I use technology to solve the system of equations?

A: Yes, you can use technology such as calculators or computer software to solve the system of equations. These tools can perform the calculations and provide the solution.

Q: What are some common mistakes to avoid when solving the system of equations?

A: Some common mistakes to avoid when solving the system of equations include:

Not following the order of operations
Not simplifying the equations
Not checking the solution
Not using the correct method to eliminate the y terms

Conclusion

In this article, we answered some frequently asked questions related to the system of equations and provided additional insights. We discussed the purpose of eliminating the y terms, how to choose the constant to eliminate the y terms, and how to use substitution or elimination to solve the system of equations. We also discussed some common mistakes to avoid when solving the system of equations.

Final Answer

The final answer is:

The purpose of eliminating the y terms is to solve for the value of x.
The choice of method to eliminate the y terms depends on the specific equation and the desired outcome.
The constant must be chosen such that it eliminates the y term when added to the second equation.
Substitution and elimination are two methods to solve the system of equations.
Technology can be used to solve the system of equations.

Note

The value of y can be found by substituting the value of x into one of the original equations. However, since our goal was to eliminate the y terms, we did not pursue this further.

References

[1] "Systems of Linear Equations" by Paul's Online Math Notes
[2] "Linear Algebra and Its Applications" by Gilbert Strang

Additional Resources

Khan Academy: Systems of Linear Equations
MIT OpenCourseWare: Linear Algebra
Wolfram Alpha: Systems of Linear Equations