For A Given Pair Of Linear Equations In Two Variables, If $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, Then The Equation Has:(A) One Solution (B) Two Solutions (C) Three Solutions (D) No Solution

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Introduction

Linear equations in two variables are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. When dealing with a pair of linear equations in two variables, it is essential to understand the concept of solutions and how to determine the number of solutions that exist. In this article, we will explore the concept of solutions for a given pair of linear equations in two variables and discuss the conditions under which the equation has one, two, three, or no solutions.

What are Linear Equations in Two Variables?

A linear equation in two variables is an equation that involves two variables, say x and y, and is expressed in the form of ax + by = c, where a, b, and c are constants. For example, the equation 2x + 3y = 5 is a linear equation in two variables. Linear equations in two variables can be graphically represented as a straight line on a coordinate plane.

Understanding the Concept of Solutions

When dealing with a pair of linear equations in two variables, we need to determine the number of solutions that exist. A solution to a linear equation is a value of the variable that satisfies the equation. In other words, it is a point on the coordinate plane that lies on the line represented by the equation. The number of solutions that exist depends on the relationship between the two equations.

Determining the Number of Solutions

To determine the number of solutions that exist, we need to compare the slopes of the two lines represented by the equations. The slope of a line is a measure of how steep the line is. If the slopes of the two lines are equal, then the lines are parallel, and there is no solution. If the slopes of the two lines are not equal, then the lines intersect at a single point, and there is one solution. If the slopes of the two lines are equal, but the y-intercepts are not equal, then the lines are coincident, and there are infinitely many solutions.

The Condition for One Solution

For a given pair of linear equations in two variables, if the ratio of the coefficients of x in the two equations is not equal to the ratio of the coefficients of y in the two equations, then the equation has one solution. This condition can be expressed mathematically as:

a1a2≠b1b2\frac{a_1}{a_2} \neq \frac{b_1}{b_2}

where a1 and b1 are the coefficients of x in the first and second equations, respectively, and a2 and b2 are the coefficients of y in the first and second equations, respectively.

The Condition for Two Solutions

For a given pair of linear equations in two variables, if the ratio of the coefficients of x in the two equations is equal to the ratio of the coefficients of y in the two equations, then the equation has two solutions. This condition can be expressed mathematically as:

a1a2=b1b2\frac{a_1}{a_2} = \frac{b_1}{b_2}

The Condition for No Solution

For a given pair of linear equations in two variables, if the ratio of the coefficients of x in the two equations is equal to the ratio of the coefficients of y in the two equations, and the y-intercepts are not equal, then the equation has no solution. This condition can be expressed mathematically as:

a1a2=b1b2 and c1≠c2\frac{a_1}{a_2} = \frac{b_1}{b_2} \text{ and } c_1 \neq c_2

where c1 and c2 are the y-intercepts of the two equations, respectively.

The Condition for Infinitely Many Solutions

For a given pair of linear equations in two variables, if the ratio of the coefficients of x in the two equations is equal to the ratio of the coefficients of y in the two equations, and the y-intercepts are equal, then the equation has infinitely many solutions. This condition can be expressed mathematically as:

a1a2=b1b2 and c1=c2\frac{a_1}{a_2} = \frac{b_1}{b_2} \text{ and } c_1 = c_2

Conclusion

In conclusion, the number of solutions that exist for a given pair of linear equations in two variables depends on the relationship between the two equations. If the ratio of the coefficients of x in the two equations is not equal to the ratio of the coefficients of y in the two equations, then the equation has one solution. If the ratio of the coefficients of x in the two equations is equal to the ratio of the coefficients of y in the two equations, then the equation has two solutions. If the ratio of the coefficients of x in the two equations is equal to the ratio of the coefficients of y in the two equations, and the y-intercepts are not equal, then the equation has no solution. If the ratio of the coefficients of x in the two equations is equal to the ratio of the coefficients of y in the two equations, and the y-intercepts are equal, then the equation has infinitely many solutions.

Example Problems

Problem 1

Solve the following system of linear equations:

2x + 3y = 5 x - 2y = -3

Solution

To solve this system of linear equations, we need to compare the slopes of the two lines represented by the equations. The slope of the first line is 2/3, and the slope of the second line is 1/2. Since the slopes are not equal, the lines intersect at a single point, and there is one solution.

Problem 2

Solve the following system of linear equations:

x + 2y = 3 2x + 4y = 6

Solution

To solve this system of linear equations, we need to compare the slopes of the two lines represented by the equations. The slope of the first line is -1/2, and the slope of the second line is -1/2. Since the slopes are equal, the lines are parallel, and there is no solution.

Problem 3

Solve the following system of linear equations:

x + 2y = 3 x + 2y = 4

Solution

To solve this system of linear equations, we need to compare the slopes of the two lines represented by the equations. The slope of the first line is -1/2, and the slope of the second line is -1/2. Since the slopes are equal, the lines are coincident, and there are infinitely many solutions.

Final Thoughts

In conclusion, the number of solutions that exist for a given pair of linear equations in two variables depends on the relationship between the two equations. By comparing the slopes of the two lines represented by the equations, we can determine the number of solutions that exist. If the slopes are not equal, then the equation has one solution. If the slopes are equal, then the equation has two solutions. If the slopes are equal, and the y-intercepts are not equal, then the equation has no solution. If the slopes are equal, and the y-intercepts are equal, then the equation has infinitely many solutions.

Q: What is the condition for one solution in a system of linear equations?

A: The condition for one solution in a system of linear equations is that the ratio of the coefficients of x in the two equations is not equal to the ratio of the coefficients of y in the two equations. This can be expressed mathematically as:

a1a2≠b1b2\frac{a_1}{a_2} \neq \frac{b_1}{b_2}

where a1 and b1 are the coefficients of x in the first and second equations, respectively, and a2 and b2 are the coefficients of y in the first and second equations, respectively.

Q: What is the condition for two solutions in a system of linear equations?

A: The condition for two solutions in a system of linear equations is that the ratio of the coefficients of x in the two equations is equal to the ratio of the coefficients of y in the two equations. This can be expressed mathematically as:

a1a2=b1b2\frac{a_1}{a_2} = \frac{b_1}{b_2}

Q: What is the condition for no solution in a system of linear equations?

A: The condition for no solution in a system of linear equations is that the ratio of the coefficients of x in the two equations is equal to the ratio of the coefficients of y in the two equations, and the y-intercepts are not equal. This can be expressed mathematically as:

a1a2=b1b2 and c1≠c2\frac{a_1}{a_2} = \frac{b_1}{b_2} \text{ and } c_1 \neq c_2

where c1 and c2 are the y-intercepts of the two equations, respectively.

Q: What is the condition for infinitely many solutions in a system of linear equations?

A: The condition for infinitely many solutions in a system of linear equations is that the ratio of the coefficients of x in the two equations is equal to the ratio of the coefficients of y in the two equations, and the y-intercepts are equal. This can be expressed mathematically as:

a1a2=b1b2 and c1=c2\frac{a_1}{a_2} = \frac{b_1}{b_2} \text{ and } c_1 = c_2

Q: How do I determine the number of solutions in a system of linear equations?

A: To determine the number of solutions in a system of linear equations, you need to compare the slopes of the two lines represented by the equations. If the slopes are not equal, then the equation has one solution. If the slopes are equal, then the equation has two solutions. If the slopes are equal, and the y-intercepts are not equal, then the equation has no solution. If the slopes are equal, and the y-intercepts are equal, then the equation has infinitely many solutions.

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

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously. A linear equation is a single equation that involves two variables. For example, the equation 2x + 3y = 5 is a linear equation, while the system of linear equations:

2x + 3y = 5 x - 2y = -3

is a system of linear equations.

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

A: To solve a system of linear equations, you need to follow these steps:

  1. Compare the slopes of the two lines represented by the equations.
  2. If the slopes are not equal, then the equation has one solution.
  3. If the slopes are equal, then the equation has two solutions.
  4. If the slopes are equal, and the y-intercepts are not equal, then the equation has no solution.
  5. If the slopes are equal, and the y-intercepts are equal, then the equation has infinitely many solutions.

Q: What is the importance of solving systems of linear equations?

A: Solving systems of linear equations is an essential skill in mathematics and is used in various fields such as physics, engineering, and economics. It is used to model real-world problems and to make predictions about the behavior of systems.

Q: Can you provide an example of a system of linear equations with one solution?

A: Yes, here is an example of a system of linear equations with one solution:

2x + 3y = 5 x - 2y = -3

To solve this system of linear equations, we need to compare the slopes of the two lines represented by the equations. The slope of the first line is 2/3, and the slope of the second line is 1/2. Since the slopes are not equal, the equation has one solution.

Q: Can you provide an example of a system of linear equations with two solutions?

A: Yes, here is an example of a system of linear equations with two solutions:

x + 2y = 3 2x + 4y = 6

To solve this system of linear equations, we need to compare the slopes of the two lines represented by the equations. The slope of the first line is -1/2, and the slope of the second line is -1/2. Since the slopes are equal, the equation has two solutions.

Q: Can you provide an example of a system of linear equations with no solution?

A: Yes, here is an example of a system of linear equations with no solution:

x + 2y = 3 x + 2y = 4

To solve this system of linear equations, we need to compare the slopes of the two lines represented by the equations. The slope of the first line is -1/2, and the slope of the second line is -1/2. Since the slopes are equal, and the y-intercepts are not equal, the equation has no solution.

Q: Can you provide an example of a system of linear equations with infinitely many solutions?

A: Yes, here is an example of a system of linear equations with infinitely many solutions:

x + 2y = 3 x + 2y = 3

To solve this system of linear equations, we need to compare the slopes of the two lines represented by the equations. The slope of the first line is -1/2, and the slope of the second line is -1/2. Since the slopes are equal, and the y-intercepts are equal, the equation has infinitely many solutions.