If 3x +5y =9 And 5x+3y =7 Then What Is The Value Of X-y.
Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will discuss how to solve a system of linear equations using the method of substitution and elimination.
The Problem
We are given two linear equations:
- 3x + 5y = 9
- 5x + 3y = 7
Our objective is to find the value of x - y.
Step 1: Write Down the Equations
Let's start by writing down the two equations:
- 3x + 5y = 9
- 5x + 3y = 7
Step 2: 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 either x or y will be the same in both equations.
Let's multiply the first equation by 5 and the second equation by 3:
- 15x + 25y = 45
- 15x + 9y = 21
Step 3: Subtract the Second Equation from the First Equation
Now, let's subtract the second equation from the first equation to eliminate the variable x:
(15x + 25y) - (15x + 9y) = 45 - 21
This simplifies to:
16y = 24
Step 4: Solve for y
Now, let's solve for y by dividing both sides of the equation by 16:
y = 24/16 y = 3/2
Step 5: Substitute the Value of y into One of the Original Equations
Now that we have the value of y, let's substitute it into one of the original equations to solve for x. We will use the first equation:
3x + 5y = 9
Substituting y = 3/2, we get:
3x + 5(3/2) = 9
Simplifying, we get:
3x + 15/2 = 9
Multiplying both sides by 2 to eliminate the fraction, we get:
6x + 15 = 18
Subtracting 15 from both sides, we get:
6x = 3
Dividing both sides by 6, we get:
x = 1/2
Step 6: Find the Value of x - y
Now that we have the values of x and y, let's find the value of x - y:
x - y = 1/2 - 3/2 x - y = -2/2 x - y = -1
Conclusion
In this article, we discussed how to solve a system of linear equations using the method of substitution and elimination. We used the given equations 3x + 5y = 9 and 5x + 3y = 7 to find the value of x - y. By following the steps outlined above, we were able to find the value of x - y, which is -1.
Tips and Tricks
- When solving a system of linear equations, it's essential to choose the correct method of substitution or elimination.
- Make sure to multiply the equations by necessary multiples to eliminate one of the variables.
- Be careful when subtracting or adding the equations to avoid errors.
- Always check your work by plugging the values back into the original equations.
Frequently Asked Questions
- What is a system of linear equations? A system of linear equations is a set of two or more linear equations that involve the same set of variables.
- How do I solve a system of linear equations? You can solve a system of linear equations using the method of substitution or elimination.
- What is the value of x - y in the given system of linear equations?
The value of x - y is -1.
Frequently Asked Questions: 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 involve the same set of variables. Each equation is a linear equation, which means it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.
Q: How do I solve a system of linear equations?
A: There are two main methods for solving a system of linear equations: substitution and elimination. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one of the variables.
Q: What is the difference between the substitution and elimination methods?
A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one of the variables. The choice of method depends on the specific system of equations and the variables involved.
Q: How do I choose the correct method for solving a system of linear equations?
A: To choose the correct method, look at the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, use the elimination method. If the coefficients of one variable are different in both equations, use the substitution method.
Q: What is the value of x - y in the given system of linear equations?
A: In the given system of linear equations 3x + 5y = 9 and 5x + 3y = 7, the value of x - y is -1.
Q: How do I check my work when solving a system of linear equations?
A: To check your work, plug the values of x and y back into the original equations. If the values satisfy both equations, then your solution is correct.
Q: What are some common mistakes to avoid when solving a system of linear equations?
A: Some common mistakes to avoid include:
- Not following the correct order of operations
- Not checking the work
- Not using the correct method for the specific system of equations
- Not being careful when adding or subtracting the equations
Q: How do I graph a system of linear equations?
A: To graph a system of linear equations, plot the equations on a coordinate plane. The point of intersection of the two lines represents the solution to the system.
Q: What is the significance of solving a system of linear equations?
A: Solving a system of linear equations has many practical applications in fields such as physics, engineering, and economics. It can be used to model real-world problems and make predictions about the behavior of systems.
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: How do I enter a system of linear equations into a calculator?
A: The steps for entering a system of linear equations into a calculator vary depending on the specific calculator model. Consult the user manual for instructions.
Q: What are some real-world applications of solving systems of linear equations?
A: Some real-world applications of solving systems of linear equations include:
- Modeling population growth
- Predicting stock prices
- Designing electrical circuits
- Solving optimization problems
Conclusion
Solving systems of linear equations is an essential skill in mathematics and has many practical applications in fields such as physics, engineering, and economics. By following the steps outlined in this article, you can solve systems of linear equations using the substitution and elimination methods. Remember to check your work and be careful when adding or subtracting the equations.