Forming Pair of Linear Equations

Oct 13, 2025 by pinkandtrash.com 135 views

Introduction

In this article, we will learn how to form a pair of linear equations from a given problem and find their solutions graphically. We will use the given problem of two numbers, where the difference between them is 26, and one number is three times the other.

Problem Statement

The problem statement is as follows:

The difference between two numbers is 26.
One number is three times the other.

Forming Pair of Linear Equations

Let's assume the two numbers are x and y. We can form the following pair of linear equations based on the given conditions:

Equation 1: Difference between two numbers is 26

Since the difference between the two numbers is 26, we can write the equation as:

x - y = 26

Equation 2: One number is three times the other

Since one number is three times the other, we can write the equation as:

x = 3y

Solving the Pair of Linear Equations

To solve the pair of linear equations, we can use the method of substitution or elimination. Let's use the method of substitution.

Substitution Method

We can substitute the value of x from Equation 2 into Equation 1:

x - y = 26

Substituting x = 3y, we get:

3y - y = 26

Simplifying the equation, we get:

2y = 26

Dividing both sides by 2, we get:

y = 13

Now that we have the value of y, we can find the value of x by substituting y into Equation 2:

x = 3y

Substituting y = 13, we get:

x = 3(13)

x = 39

Therefore, the two numbers are 39 and 13.

Graphical Representation

To represent the pair of linear equations graphically, we can plot the lines on a coordinate plane.

Equation 1: x - y = 26

We can rewrite the equation as:

y = x - 26

This is a linear equation in slope-intercept form, where the slope is 1 and the y-intercept is -26.

Equation 2: x = 3y

We can rewrite the equation as:

y = x/3

This is a linear equation in slope-intercept form, where the slope is 1/3 and the y-intercept is 0.

Plotting the Lines

To plot the lines, we can use the slope-intercept form of the equations.

Line 1: y = x - 26

We can plot the line by finding two points on the line. Let's find the points where x = 0 and x = 26.

When x = 0, y = -26. So, the point (0, -26) lies on the line.

When x = 26, y = 0. So, the point (26, 0) lies on the line.

We can plot the line by drawing a straight line through the points (0, -26) and (26, 0).

Line 2: y = x/3

We can plot the line by finding two points on the line. Let's find the points where x = 0 and x = 3.

When x = 0, y = 0. So, the point (0, 0) lies on the line.

When x = 3, y = 1. So, the point (3, 1) lies on the line.

We can plot the line by drawing a straight line through the points (0, 0) and (3, 1).

Finding the Intersection Point

To find the intersection point of the two lines, we can find the point where the lines intersect.

We can see that the lines intersect at the point (39, 13).

Therefore, the two numbers are 39 and 13.

Conclusion

In this article, we learned how to form a pair of linear equations from a given problem and find their solutions graphically. We used the method of substitution to solve the pair of linear equations and found the intersection point of the two lines. We also plotted the lines on a coordinate plane and found the point where the lines intersect.

Key Takeaways

Forming pair of linear equations from a given problem
Solving pair of linear equations using substitution method
Graphical representation of pair of linear equations
Finding intersection point of two lines

Practice Problems

The difference between two numbers is 15, and one number is twice the other. Find the numbers.
The sum of two numbers is 20, and one number is three times the other. Find the numbers.
The difference between two numbers is 25, and one number is four times the other. Find the numbers.

Solutions

Let's assume the two numbers are x and y. We can form the following pair of linear equations based on the given conditions:

x - y = 15

x = 2y

Substituting x = 2y into the first equation, we get:

2y - y = 15

Simplifying the equation, we get:

y = 15

Now that we have the value of y, we can find the value of x by substituting y into the second equation:

x = 2y

Substituting y = 15, we get:

x = 2(15)

x = 30

Therefore, the two numbers are 30 and 15.
Let's assume the two numbers are x and y. We can form the following pair of linear equations based on the given conditions:

x + y = 20

x = 3y

Substituting x = 3y into the first equation, we get:

3y + y = 20

Simplifying the equation, we get:

4y = 20

Dividing both sides by 4, we get:

y = 5

Now that we have the value of y, we can find the value of x by substituting y into the second equation:

x = 3y

Substituting y = 5, we get:

x = 3(5)

x = 15

Therefore, the two numbers are 15 and 5.
Let's assume the two numbers are x and y. We can form the following pair of linear equations based on the given conditions:

x - y = 25

x = 4y

Substituting x = 4y into the first equation, we get:

4y - y = 25

Simplifying the equation, we get:

3y = 25

Dividing both sides by 3, we get:

y = 25/3

Now that we have the value of y, we can find the value of x by substituting y into the second equation:

x = 4y

Substituting y = 25/3, we get:

x = 4(25/3)

x = 100/3

Therefore, the two numbers are 100/3 and 25/3.
Q&A: Pair of Linear Equations =====================================

Frequently Asked Questions

In this article, we will answer some frequently asked questions about pair of linear equations.

Q1: What is a pair of linear equations?

A pair of linear equations is a system of two linear equations in two variables. It is a set of two equations that can be solved simultaneously to find the values of the variables.

Q2: How do I form a pair of linear equations from a given problem?

To form a pair of linear equations from a given problem, you need to identify the variables and the relationships between them. You can then write two equations based on the given conditions.

Q3: What are the different methods to solve a pair of linear equations?

There are two main methods to solve a pair of linear equations: substitution method and elimination method.

Substitution Method: In this method, you substitute the value of one variable from one equation into the other equation.
Elimination Method: In this method, you add or subtract the two equations to eliminate one variable.

Q4: How do I find the intersection point of two lines?

To find the intersection point of two lines, you need to solve the pair of linear equations simultaneously. You can use either the substitution method or the elimination method to find the values of the variables.

Q5: What is the graphical representation of a pair of linear equations?

The graphical representation of a pair of linear equations is a plot of the two lines on a coordinate plane. The intersection point of the two lines represents the solution to the pair of linear equations.

Q6: How do I plot the lines on a coordinate plane?

To plot the lines on a coordinate plane, you need to find two points on each line. You can then draw a straight line through the points to represent the line.

Q7: What are the key takeaways from solving a pair of linear equations?

The key takeaways from solving a pair of linear equations are:

Forming pair of linear equations from a given problem
Solving pair of linear equations using substitution method or elimination method
Graphical representation of pair of linear equations
Finding intersection point of two lines

Q8: What are some practice problems to solve?

Here are some practice problems to solve:

The difference between two numbers is 15, and one number is twice the other. Find the numbers.
The sum of two numbers is 20, and one number is three times the other. Find the numbers.
The difference between two numbers is 25, and one number is four times the other. Find the numbers.

Q9: How do I check my solutions?

To check your solutions, you need to substitute the values of the variables back into the original equations. If the equations are true, then your solutions are correct.

Q10: What are some common mistakes to avoid?

Here are some common mistakes to avoid when solving a pair of linear equations:

Not identifying the variables and the relationships between them
Not using the correct method to solve the equations
Not checking the solutions
Not plotting the lines on a coordinate plane

Conclusion

In this article, we have answered some frequently asked questions about pair of linear equations. We have covered the basics of forming pair of linear equations, solving pair of linear equations using substitution method and elimination method, graphical representation of pair of linear equations, and finding intersection point of two lines. We have also provided some practice problems to solve and some common mistakes to avoid.