Understanding the Pair of Equations

Oct 13, 2025 by pynkocean.com 81 views

Solving the Pair of Equations: Unraveling the Mystery of x=a and y=b

In the realm of mathematics, solving a pair of linear equations is a fundamental concept that forms the basis of algebra. The given problem involves finding the values of a and b, where x=a and y=b is the solution of the pair of equations x-y=2 and x+y=4. In this article, we will delve into the world of linear equations, explore the concept of solutions, and unravel the mystery of finding the values of a and b.

Understanding the Pair of Equations

A pair of linear equations in two variables is a set of two equations, each containing two variables, x and y. The given pair of equations is:

x - y = 2 ... (1) x + y = 4 ... (2)

The Concept of Solutions

A solution to a pair of linear equations is a value of x and y that satisfies both equations. In other words, if we substitute the values of x and y into both equations, the equations should hold true. In this case, we are given that x=a and y=b is the solution of the pair of equations.

Substituting x=a and y=b into the Pair of Equations

To find the values of a and b, we will substitute x=a and y=b into both equations (1) and (2).

Substituting x=a and y=b into equation (1):

a - b = 2 ... (3)

Substituting x=a and y=b into equation (2):

a + b = 4 ... (4)

Solving the System of Equations

We now have a system of two equations with two variables. To solve for a and b, we can use the method of substitution or elimination. In this case, we will use the elimination method.

Elimination Method

We will multiply equation (3) by 1 and equation (4) by 1, and then add both equations to eliminate the variable b.

(a - b) + (a + b) = 2 + 4 2a = 6

Solving for a

Now that we have eliminated the variable b, we can solve for a.

a = 6/2 a = 3

Substituting a=3 into Equation (4)

Now that we have found the value of a, we can substitute it into equation (4) to find the value of b.

3 + b = 4 b = 4 - 3 b = 1

In this article, we have solved the pair of equations x-y=2 and x+y=4, where x=a and y=b is the solution. We have used the elimination method to find the values of a and b, and have found that a=3 and b=1. This problem has helped us understand the concept of solutions to a pair of linear equations and has provided us with a clear understanding of how to solve such problems.

Key Takeaways

A pair of linear equations in two variables is a set of two equations, each containing two variables, x and y.
A solution to a pair of linear equations is a value of x and y that satisfies both equations.
The elimination method can be used to solve a system of two equations with two variables.
By substituting x=a and y=b into both equations, we can find the values of a and b.

Practice Problems

Solve the pair of equations x+y=3 and x-y=1, where x=a and y=b is the solution.
Solve the pair of equations 2x+y=5 and x-2y=-3, where x=a and y=b is the solution.

References

Glossary

Linear Equation: An equation in which the highest power of the variable(s) is 1.
Pair of Linear Equations: A set of two equations, each containing two variables, x and y.
Solution: A value of x and y that satisfies both equations.
Elimination Method: A method of solving a system of two equations with two variables by adding or subtracting the equations to eliminate one of the variables.
Frequently Asked Questions (FAQs) on Solving the Pair of Equations: x-y=2 and x+y=4

Q: What is the solution to the pair of equations x-y=2 and x+y=4?

A: The solution to the pair of equations x-y=2 and x+y=4 is x=3 and y=1.

Q: How did you find the values of x and y?

A: We used the elimination method to solve the system of two equations with two variables. We multiplied equation (3) by 1 and equation (4) by 1, and then added both equations to eliminate the variable b. This gave us the value of a, which was 3. We then substituted a=3 into equation (4) to find the value of b, which was 1.

Q: What is the elimination method?

A: The elimination method is a method of solving a system of two equations with two variables by adding or subtracting the equations to eliminate one of the variables.

Q: Can you explain the concept of solutions to a pair of linear equations?

A: A solution to a pair of linear equations is a value of x and y that satisfies both equations. In other words, if we substitute the values of x and y into both equations, the equations should hold true.

Q: How do you know which method to use to solve a system of two equations with two variables?

A: There are several methods that can be used to solve a system of two equations with two variables, including the substitution method, the elimination method, and the graphing method. The choice of method depends on the specific equations and the values of the variables.

Q: What is the difference between the substitution method and the elimination method?

A: The substitution method involves substituting one equation into the other equation to solve for one of the variables. The elimination method involves adding or subtracting the equations to eliminate one of the variables.

Q: Can you provide an example of how to use the substitution method to solve a system of two equations with two variables?

A: Yes, here is an example:

Suppose we have the following system of two equations with two variables:

x + y = 3 2x - y = 5

We can use the substitution method to solve for x and y. First, we will solve the first equation for y:

y = 3 - x

Next, we will substitute this expression for y into the second equation:

2x - (3 - x) = 5

Simplifying this equation, we get:

2x - 3 + x = 5

Combine like terms:

3x - 3 = 5

Add 3 to both sides:

3x = 8

Divide both sides by 3:

x = 8/3

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:

x + y = 3

Substitute x = 8/3:

8/3 + y = 3

Subtract 8/3 from both sides:

y = 3 - 8/3

Simplify this expression:

y = (9 - 8)/3

y = 1/3

Therefore, the solution to the system of two equations with two variables is x = 8/3 and y = 1/3.

Q: Can you provide an example of how to use the graphing method to solve a system of two equations with two variables?

A: Yes, here is an example:

Suppose we have the following system of two equations with two variables:

x + y = 3 2x - y = 5

We can use the graphing method to solve for x and y. First, we will graph the two equations on a coordinate plane. The first equation is a line with a slope of 1 and a y-intercept of 3. The second equation is a line with a slope of 2 and a y-intercept of -5.

To find the solution to the system of two equations with two variables, we need to find the point of intersection between the two lines. This point of intersection represents the solution to the system of two equations with two variables.

In this case, the point of intersection is (2, 1). Therefore, the solution to the system of two equations with two variables is x = 2 and y = 1.

Q: What are some common mistakes to avoid when solving a system of two equations with two variables?

A: Some common mistakes to avoid when solving a system of two equations with two variables include:

Not checking to see if the equations are linear
Not checking to see if the equations are consistent
Not using the correct method to solve the system of two equations with two variables
Not checking to see if the solution is a valid solution

Q: How do you know if a system of two equations with two variables has a solution?

A: A system of two equations with two variables has a solution if the equations are consistent and the solution is a valid solution. If the equations are inconsistent, then the system of two equations with two variables does not have a solution.

Q: Can you provide an example of how to determine if a system of two equations with two variables has a solution?

A: Yes, here is an example:

Suppose we have the following system of two equations with two variables:

x + y = 3 x + y = 4

We can determine if this system of two equations with two variables has a solution by checking to see if the equations are consistent. If the equations are consistent, then the system of two equations with two variables has a solution. If the equations are inconsistent, then the system of two equations with two variables does not have a solution.

In this case, the two equations are identical, so they are consistent. Therefore, the system of two equations with two variables has a solution.

However, if we had the following system of two equations with two variables:

x + y = 3 x - y = 4

In this case, the two equations are inconsistent, so the system of two equations with two variables does not have a solution.

Q: Can you provide an example of how to use the substitution method to solve a system of two equations with two variables that has no solution?

A: Yes, here is an example:

Suppose we have the following system of two equations with two variables:

x + y = 3 x - y = 4

We can use the substitution method to solve for x and y. First, we will solve the first equation for y:

y = 3 - x

Next, we will substitute this expression for y into the second equation:

x - (3 - x) = 4

Simplifying this equation, we get:

x - 3 + x = 4

Combine like terms:

2x - 3 = 4

Add 3 to both sides:

2x = 7

Divide both sides by 2:

x = 7/2

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:

x + y = 3

Substitute x = 7/2:

7/2 + y = 3

Subtract 7/2 from both sides:

y = 3 - 7/2

Simplify this expression:

y = (6 - 7)/2

y = -1/2

Therefore, the solution to the system of two equations with two variables is x = 7/2 and y = -1/2.

However, if we check to see if the solution x = 7/2 and y = -1/2 satisfies both equations, we find that it does not. Therefore, the system of two equations with two variables has no solution.

Q: Can you provide an example of how to use the graphing method to solve a system of two equations with two variables that has no solution?

A: Yes, here is an example:

Suppose we have the following system of two equations with two variables:

x + y = 3 x - y = 4

To find the solution to the system of two equations with two variables, we need to find the point of intersection between the two lines. However, in this case, the two lines are parallel, so they do not intersect. Therefore, the system of two equations with two variables has no solution.

Q: What are some common mistakes to avoid when graphing a system of two equations with two variables?

A: Some common mistakes to avoid when graphing a system of two equations with two variables include:

Not checking to see if the equations are linear
Not checking to see if the equations are consistent
Not using the correct method to graph the system of two equations with two variables