−9x+8y= −26 Minus, 9, X, Minus, 6, Y, Equals, Minus, 12 −9x−6y= −12
Introduction
Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving two linear equations: −9x+8y= −26 and −9x−6y= −12. We will use a step-by-step approach to solve these equations and provide a clear understanding of the concepts involved.
What are Linear Equations?
A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables. Linear equations can be solved using various methods, including substitution, elimination, and graphing.
The Given Equations
We are given two linear equations:
- −9x+8y= −26
- −9x−6y= −12
Step 1: Write Down the Equations
To solve these equations, we need to write them down in a clear and concise manner.
Equation 1: −9x+8y= −26
-9x + 8y = -26
Equation 2: −9x−6y= −12
-9x - 6y = -12
Step 2: Identify the Coefficients
To solve these equations, we need to identify the coefficients of x and y in each equation.
Equation 1: −9x+8y= −26
- Coefficient of x: -9
- Coefficient of y: 8
Equation 2: −9x−6y= −12
- Coefficient of x: -9
- Coefficient of y: -6
Step 3: Multiply the Equations
To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of one of the variables (in this case, x) are the same in both equations.
Multiply Equation 1 by 1 and Equation 2 by 1
-9x + 8y = -26
-9x - 6y = -12
Step 4: Add the Equations
Now that we have the same coefficients for x in both equations, we can add the equations to eliminate x.
(-9x + 8y) + (-9x - 6y) = -26 + (-12)
-18x + 2y = -38
Step 5: Solve for y
Now that we have eliminated x, we can solve for y.
-18x + 2y = -38
2y = -38 + 18x
y = (-38 + 18x)/2
Step 6: Substitute y into One of the Original Equations
Now that we have the value of y, we can substitute it into one of the original equations to solve for x.
Substitute y into Equation 1
-9x + 8y = -26
-9x + 8((-38 + 18x)/2) = -26
Step 7: Solve for x
Now that we have substituted y into one of the original equations, we can solve for x.
-9x + 8((-38 + 18x)/2) = -26
-9x + 4(-38 + 18x) = -26
-9x - 152 + 72x = -26
63x = 126
x = 126/63
x = 2
Step 8: Find the Value of y
Now that we have the value of x, we can find the value of y.
y = (-38 + 18x)/2
y = (-38 + 18(2))/2
y = (-38 + 36)/2
y = -2/2
y = -1
Conclusion
In this article, we have solved two linear equations: −9x+8y= −26 and −9x−6y= −12. We used a step-by-step approach to eliminate one of the variables and solve for the other variable. We also provided a clear understanding of the concepts involved in solving linear equations.
Final Answer
Introduction
In our previous article, we solved two linear equations: −9x+8y= −26 and −9x−6y= −12. In this article, we will provide a Q&A guide to help you understand the concepts involved in solving linear equations.
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.
Q: What are the different methods for solving linear equations?
A: There are several methods for solving linear equations, including:
- Substitution method
- Elimination method
- Graphing method
Q: What is the substitution method?
A: The substitution method involves substituting the value of one variable into the other equation to solve for the other variable.
Q: What is the elimination method?
A: The elimination method involves eliminating one of the variables by adding or subtracting the equations.
Q: What is the graphing method?
A: The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection.
Q: How do I choose which method to use?
A: The choice of method depends on the type of equation and the variables involved. If the equations are simple and have the same coefficients, the substitution method may be the easiest. If the equations have different coefficients, the elimination method may be more effective.
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Not checking the work
- Not using the correct method
- Not simplifying the equations
- Not checking for extraneous solutions
Q: How do I check my work when solving linear equations?
A: To check your work, you can:
- Plug the solution back into the original equations
- Check that the solution satisfies both equations
- Use a calculator or computer program to verify the solution
Q: What is an extraneous solution?
A: An extraneous solution is a solution that is not valid or is not a solution to the original equation.
Q: How do I avoid extraneous solutions?
A: To avoid extraneous solutions, you can:
- Check the work carefully
- Use the correct method
- Simplify the equations
- Check for extraneous solutions
Q: Can I use linear equations to solve real-world problems?
A: Yes, linear equations can be used to solve real-world problems. For example, you can use linear equations to model the cost of goods, the time it takes to complete a task, or the amount of money in a bank account.
Q: What are some examples of real-world problems that can be solved using linear equations?
A: Some examples of real-world problems that can be solved using linear equations include:
- Finding the cost of goods
- Determining the time it takes to complete a task
- Calculating the amount of money in a bank account
- Modeling the growth of a population
- Determining the amount of fuel needed for a trip
Conclusion
In this article, we have provided a Q&A guide to help you understand the concepts involved in solving linear equations. We have covered topics such as the different methods for solving linear equations, common mistakes to avoid, and how to check your work. We have also provided examples of real-world problems that can be solved using linear equations.
Final Answer
The final answer is that linear equations are a powerful tool for solving real-world problems. By understanding the concepts involved in solving linear equations, you can apply them to a wide range of problems and make informed decisions.