Linear equations with two unknowns are a common topic in mathematics. These equations are used to find the relationship between two variables. In this article, we will discuss what linear equations with two unknowns are, how to solve them, and their applications.
What are Linear Equations with Two Unknowns?
A linear equation with two unknowns is an equation in which two variables are involved, and both the variables are raised to the first power. These equations are also called simultaneous linear equations or simply linear equations.
The general form of a linear equation with two unknowns is:
ax + by = c
Here, x and y are the variables, a, b, and c are constants, and a and b are not equal to zero.
How to Solve Linear Equations with Two Unknowns?
The simplest method to solve linear equations with two unknowns is by substitution. In this method, you solve one equation for one variable and substitute it into the other equation. The steps to solve a linear equation with two unknowns are:
- Write down the two equations in the standard form.
- Solve one equation for one variable.
- Substitute the expression from step 2 into the second equation.
- Solve the resulting equation for the remaining variable.
- Substitute the value of the remaining variable into one of the original equations and solve for the other variable.
Let's take an example to understand this method:
2x + 3y = 12 ... (1)
x - y = 1 ... (2)
From equation (2), we get:
x = y + 1
Substituting x = y + 1 in equation (1), we get:
2(y + 1) + 3y = 12
2y + 2 + 3y = 12
5y = 10
y = 2
Substituting y = 2 in equation (2), we get:
x - 2 = 1
x = 3
Therefore, the solution of the given system of equations is x = 3 and y = 2.
Applications of Linear Equations with Two Unknowns
Linear equations with two unknowns are used in various fields such as engineering, physics, and economics. Some of the applications of linear equations with two unknowns are:
- Optimization problems in business and economics
- Calculating break-even points
- Calculating the intersection point of two lines
- Calculating the slope of a line
Linear equations with two unknowns are also used in real-life situations, such as calculating the amount of ingredients required for a recipe or calculating the distance between two cities.
Conclusion
Linear equations with two unknowns are an essential topic in mathematics. They are used to find the relationship between two variables and are applied in various fields such as engineering, physics, and economics. By understanding how to solve linear equations with two unknowns, you can solve real-life problems and make informed decisions.
