Solving Systems of Linear Equations in Two Variables

Applications of Linear Equations in Two Variables

A linear equation in two variables is a type of equation that represents a relationship between two variables, typically expressed in the form of ax + by + c = 0, where a and b are not equal to 0. Linear equations in two variables are fundamental in various fields of study, including physics, economics, engineering, and computer science. They have numerous practical applications, such as modeling real-life situations, calculating costs, and determining relationships between physical quantities.

Solving Systems of Linear Equations in Two Variables

In linear algebra, a system of linear equations involves two or more equations with two or more variables each. Given two linear equations in two variables, the system is solvable if the slopes of the corresponding lines are different, meaning nonparallel lines intersect at exactly one point. This intersection point represents the unique solution to the system of equations. When solving a system of linear equations, graphical methods, such as the substitution method, can be applied to determine the solution.

Substitution Method

The substitution method allows you to solve a system of linear equations by using one equation to solve for one of the variables, then replacing that variable with its value in the other equation. This leads to a single linear equation with one variable, which can be easily solved.

Example

For instance, let's consider the system of equations:

2x + 5y = 20
3x + 6y = 12

To apply the substitution method, we first isolate x in the first equation:

2x + 5y = 20
2x = 20 - 5y
x = (20 - 5y) / 2

Now, substitute x from the first equation back into the second equation:

3x + 6y = 12
3((20 - 5y) / 2) + 6y = 12

Simplifying this equation:

3(20 - 5y) / 2 + 6y = 12

Expanding and simplifying further:

3(20 - 5y) / 2 + 6y = 12
3(20 - 5y) = 12 * 2
3(20 - 5y) = 24

Now, we can solve for y:

3(20 - 5y) = 24
20 - 5y = 8
5y = 12
y = 12 / 5

Finally, substitute the value of y back into the equation x = (20 - 5y) / 2 to find x:

x = (20 - 5(12 / 5)) / 2
x = (20 - 12) / 2
x = 8 / 2
x = 4

So, the solution to the given system of linear equations is (x, y) = (4, 12).

Graphing Linear Equations

Graphing linear equations provides a visual representation of the relationship between the variables. Each line on a coordinate plane corresponds to a single linear equation. When two lines representing distinct equations intersect, the coordinates of their point of intersection represent the unique solution to the system of equations. If the two lines are parallel or coincident, meaning they have the same slope, the system has no solution.