Solving a System of Linear Equations

Introduction 

A linear equation is one in which the variable’s maximum power is always 1. A one-degree equation is another name for it. Ax + B = 0 is the conventional form of a linear equation in one variable. A & B are constants, and x is a variable. A linear equation with two variables is written in the conventional form Ax + By = C. A, B, & C are constants, while x and y are variables.

Definition of a Linear Equation

An equation with a degree of one is known as a linear equation. This signifies that no variable in a linear equation has an exponent bigger than 1. A linear equation’s graph is always a straight line.

Forms of Linear Equation

The three types of linear equations are as follows:

• Standard Form
• Slope Intercept Form
• Point Slope Form

A linear equation is expressed using the Linear Equation Formula. This can be accomplished in a variety of ways. A linear equation can be written in standard form, slope-intercept form, or point-slope form, for example. Let’s look at the usual form of a linear equation and see how it’s written. It fluctuates depending on the number of variables, and it’s important to remember that all variables in the equation should have the highest (and only) degree of 1.

System of Equations

Simultaneous equations, a system of equations two or more equations must be solved jointly in algebra. The number of equations must equal the number of unknowns for a system to have a unique solution. Even so, there is no certainty that a solution will be found. The system is consistent if there is a solution; otherwise, it is inconsistent. A matrix with the coefficients of the equations can be used to describe a system of linear equations. Though small two-equation systems with two unknowns can be solved using substitution, larger systems are best solved using matrix approaches.

Solving a System of Linear Equations

• Linear Equations with One Variable Solution

In order to solve a linear equation, both sides of the equation must be balanced. The equality sign indicates that both sides of the ‘equal to’ sign are equal. Because the equation is balanced, specific mathematical operations are done on both sides of the equation so that the equation’s balance is preserved.

• Linear Equations in Two Variables Solution

There are several ways for solving linear equations in two variables. Some of them are listed below. Substitution, elimination, and graphing are the three methods for solving systems of linear equations or you can use systems of equations calculators.

Substitution

In one of the equations, get a variable by itself.

• Take the expression you received for the variable in step 1 and enter it into the other equation.
• Solve the equation for the remaining variable.
• The result should be plugged into the equation from step 1’s equation.

Elimination

• Rearrange both equations, if necessary, so that the xx-terms come first, then the yy-terms, the equals sign, and the constant term. If an equation does not appear to have a constant term, the constant term is 00.
• Multiply one (or both) equations by a constant that allows the xx- or y-terms to cancel out when the equations are added or subtracted.
• Subtract or add the equations.
• Calculate the value of the last variable.
• Step 4’s answer should be plugged into one of the original equations, and the other variable should be solved.

Graphing

• In each equation, solve for y.
• Both equations should be graphed in the same Cartesian coordinate system.
• Find the location where the lines cross (the point where the lines cross).

Solving a System of Linear Equations by Substitution

Substitution is used to solve a system of equations.

One of the two variables in one of the equations should be isolated.

In the other equation, substitute the expression equal to the isolated variable from Step 1. This should result in a single-variable linear equation.

For the remaining variable, solve the linear equation.

Using one of the original equations, compute the value of the other variable in the system using the solution from Step 3.

The process of reducing two equations with two variables to a single equation with one variable is known as solving a system of equations. Because each equation in the system includes two variables, substituting an expression for a variable is one technique to reduce the number of variables in an equation.

Consider the following illustration:

X = 2y

X + y = 3

Both equations are true at the same time in a system of equations. In other words, because x is equal to 2y in the first equation, x in the second equation is also equal to 2y. As a result, we can swap 2y for x in the second equation:

=> x + y = 3

=> 2y + y = 3

=> 3y = 3

=> y = 1

 then compute x using the value of y. 

=> x = 2

Method of Elimination

What would be a good beginning step in solving the system by elimination?

X+3y =12

2x-y = 5

When we solve a system via elimination, we are getting rid of (eliminating) one of the variables. As a result, we must be able to add or subtract the equations, canceling either the x-terms or the y-terms in the process.

Any of the following perspectives would be a good place to start:

• Multiply the first equation by -2 or 2 to get the second equation. In both equations, this will result in -2x or 2x, causing the x-terms to cancel when we add or subtract.
• Multiply the second equation by 3 or -3  times. In both equations, this will result in 3y or -3y, causing the y-terms to cancel when we add or subtract.
• Divide 2 from the second equation. In both equations, this will result in x or -x, causing the x-terms to cancel when we add or remove.
• Divide the first equation by 3. This will result in y or -y in both equations, causing the y-terms to cancel when added or subtracted.

Conclusion

Thus a system of linear equations is made up of two or more equations, each of which has two or more variables, and all of the equations in the system are considered simultaneously.

Any ordered pair that satisfies each equation separately is the solution to a system of linear equations in two variables.

Independent systems of equations have one solution, dependent systems have an infinite number of solutions, and incompatible systems have no answer.

Real-world problems with multiple variables, such as those involving revenue, cost, and profit, can be solved using systems of equations.