Solve Each System. Explain Your Choiceofsolution Method28.6x-5y=-16x+4y=-10

Feb 26, 2025 by ADMIN 76 views

Solving Systems of Linear Equations: A Step-by-Step Guide

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Introduction

Solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the solution to a set of two or more linear equations with two or more variables. In this article, we will focus on solving a system of two linear equations with two variables using the method of substitution and elimination.

The System of Linear Equations

The given system of linear equations is:

2.8x - 5y = -16
-4x + 4y = -10

Choosing a Solution Method

There are several methods to solve a system of linear equations, including:

Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
Elimination Method: This method involves adding or subtracting the equations to eliminate one variable.
Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

In this case, we will use the Elimination Method to solve the system of linear equations.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one variable, we need to multiply the equations by necessary multiples such that the coefficients of one variable (either x or y) are the same in both equations.

Let's multiply the first equation by 4 and the second equation by 2.5:

11.2x - 20y = -64
-10x + 10y = -25

Step 2: Add the Equations

Now, we can add the two equations to eliminate the variable x:

(11.2x - 20y) + (-10x + 10y) = -64 + (-25)
1.2x - 10y = -89

Step 3: Solve for y

Next, we can solve for y by isolating it on one side of the equation:

-10y = -89 - 1.2x
y = (89 + 1.2x) / 10

Step 4: Substitute y into One of the Original Equations

Now, we can substitute the expression for y into one of the original equations to solve for x. Let's substitute it into the first equation:

2.8x - 5((89 + 1.2x) / 10) = -16

Step 5: Simplify and Solve for x

Simplifying the equation, we get:

2.8x - (89 + 1.2x) / 2 = -16
2.8x - 44.5 - 0.6x = -16
2.2x = 28.5
x = 13

Step 6: Find the Value of y

Now that we have the value of x, we can find the value of y by substituting it into the expression we derived earlier:

y = (89 + 1.2(13)) / 10
y = (89 + 15.6) / 10
y = 104.6 / 10
y = 10.46

Conclusion

In this article, we solved a system of two linear equations with two variables using the elimination method. We multiplied the equations by necessary multiples, added them to eliminate one variable, and then solved for the other variable. The solution to the system is x = 13 and y = 10.46.

Frequently Asked Questions

Q: What is the elimination method?

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

Q: How do I choose a solution method?

A: You can choose a solution method based on the type of equations and the variables involved. The substitution method is useful when one equation is already solved for one variable, while the elimination method is useful when the coefficients of one variable are the same in both equations.

Q: What is the graphical method?

A: The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection. This method is useful for visualizing the solution to a system of linear equations.

Final Thoughts

Solving systems of linear equations is an essential skill in mathematics, particularly in algebra and geometry. By choosing the right solution method and following the steps outlined in this article, you can solve systems of linear equations with ease. Remember to always check your work and verify the solution to ensure accuracy.

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Introduction

Solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. In our previous article, we discussed the elimination method for solving systems of linear equations. In this article, we will provide a Q&A guide to help you better understand the concept and solve systems of linear equations.

Q&A Guide

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations with two or more variables. It involves finding the solution to a set of equations where each equation is in the form of ax + by = c.

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

A: There are three main methods for solving systems of linear equations:

Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
Elimination Method: This method involves adding or subtracting the equations to eliminate one variable.
Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: How do I choose a solution method?

Q: What is the elimination method?

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

Q: How do I multiply the equations by necessary multiples?

A: To multiply the equations by necessary multiples, you need to multiply both sides of each equation by a number that will make the coefficients of one variable the same in both equations.

Q: What is the graphical method?

Q: How do I graph the equations on a coordinate plane?

A: To graph the equations on a coordinate plane, you need to plot the x and y intercepts of each equation. The x-intercept is the point where the equation crosses the x-axis, and the y-intercept is the point where the equation crosses the y-axis.

Q: What is the point of intersection?

A: The point of intersection is the point where the two lines intersect. This is the solution to the system of linear equations.

Q: How do I check my work?

A: To check your work, you need to substitute the solution back into both original equations to make sure it satisfies both equations.

Q: What are some common mistakes to avoid when solving systems of linear equations?

A: Some common mistakes to avoid when solving systems of linear equations include:

Not checking your work
Not following the correct order of operations
Not using the correct solution method
Not simplifying the equations

Conclusion

Solving systems of linear equations is an essential skill in mathematics, particularly in algebra and geometry. By understanding the different methods for solving systems of linear equations and following the steps outlined in this article, you can solve systems of linear equations with ease. Remember to always check your work and verify the solution to ensure accuracy.

Frequently Asked Questions

Q: What is the difference between a system of linear equations and a system of nonlinear equations?

A: A system of linear equations is a set of linear equations with two or more variables, while a system of nonlinear equations is a set of nonlinear equations with two or more variables.

Q: Can I use the elimination method to solve a system of nonlinear equations?

A: No, the elimination method is only used to solve systems of linear equations. To solve a system of nonlinear equations, you need to use a different method, such as the substitution method or the graphical method.

Q: How do I solve a system of linear equations with three variables?

A: To solve a system of linear equations with three variables, you need to use the elimination method or the substitution method. You can also use the graphical method to visualize the solution.