Solving Linear Equations: Substitution, Elimination, Cross Multiplication, Graphing

Solving Linear Equations

Linear equations are mathematical expressions where every term is either of degree one or zero. They have many applications in various fields such as physics, engineering, computer science, economics, social sciences, and more. Solving linear equations involves finding the values of variables that make the equation true. There are several methods to solve linear equations, including substitution, elimination, cross multiplication, and graphing. In this article, we will discuss these methods in detail.

Substitution Method

The substitution method is used when dealing with two-step equations. It involves solving the first equation for one variable while keeping the other constant, and then using the resulting value to substitute back into the second equation to find the value of the remaining variable. For example, consider the system of equations:

x + y = 5
2x - y = 6

To solve this system using the substitution method, you would first solve the first equation for x:

x + y = 5
y = 5 - x

Now, take this expression and substitute it back into the second equation:

2(5 - x) - y = 6
2(5 - x) - (5 - x) = 6

Simplifying and solving for x:

2 \* 5 - 2 \* x - 5 + x = 6
10 - 2x + x = 6
2x = 4
x = 2

Now that we have the value of x, we can plug it back into the first equation to find the value of y:

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

Elimination Method

The elimination method is used when dealing with two-step equations. It involves adding or subtracting the equations to eliminate one of the variables. For example, consider the system of equations:

x + y = 5
2x + y = 6

To solve this system using the elimination method, you would first subtract the first equation from the second equation:

2x + y = 6
2x + y - (x + y) = 6

Simplifying and solving for x:

2x + y - x - y = 6
x = 6

Now that we have the value of x, we can plug it back into either equation to find the value of y:

x + y = 5
6 + y = 5
y = 5 - 6 = -1

Cross Multiplication Method

The cross multiplication method is used when dealing with two equations with two variables. It involves cross-multiplying the equations and then solving for one of the variables. For example, consider the system of equations:

x/2 = y/3
x/4 = y/5

To solve this system using the cross multiplication method, you would first cross-multiply:

x/2 = y/3
2y = 3x

x/4 = y/5
5x = 4y

Now, we have two equations with two variables. We can solve one of them for one of the variables, and then plug that value into the other equation to find the value of the remaining variable. Let's solve the first equation for x:

2y = 3x
x = (2y)/3

Now, plug this expression for x into the second equation:

5((2y)/3) = 4y
2y = 12y/3

Simplifying and solving for y:

10y/3 = 12y/3
y = 12y/10
y = 12/10
y = 1.2

Now that we have the value of y, we can plug it back into either equation to find the value of x:

x/2 = y/3
x/2 = (1.2)/3
x = 2 \* (1.2)/3
x = 2 \* (4/15)
x = 8/15

Graphing Method

The graphing method is used when dealing with two-step equations. It involves graphing the equations and finding the intersection point, which represents the solution. For example, consider the system of equations:

x + y = 5
x - y = 3

To solve this system using the graphing method, you would first graph each equation on the same coordinate plane:

y = x + 5
y = x - 3

The equations intersect at the point (2, 2). This means that when x is 2 and y is 2, both equations are true. Therefore, the solution to the system of equations is x = 2 and y = 2.

In conclusion, solving linear equations involves various methods such as substitution, elimination, cross multiplication, and graphing. Each method has its own advantages and limitations, and choosing the appropriate method depends on the specific problem at hand.