Systems of Equation in Two Variables

A system of linear equations is made up of two or more linear equations made up of two or more variables that are all considered at the same time. It is written in the form

ax+by=p

cx+dy=q

We must discover a numerical value for each variable in the system that will satisfy all of the system’s equations at the same time to find the unique solution to a system of linear equations. There may be no solution for some linear systems, while others may have an endless number of solutions. There must be at least as many equations as variables in a linear system for it to have a unique solution. 

How do we write a System of Linear Equation Given Situation 

We write the given situation that represents a system of linear equations and find its solution as follows:

1. Determine each linear model’s input and output.
2. Determine each linear model’s slope and y-intercept.
3. Set the two linear functions equal to one another and solve for x, or find the point of intersection on a graph to obtain the solution.

Types of Linear Systems

There exist three types of linear equations in two variables, and thus three types of solutions.

• In the independent system, there exists only one solution pair (x,y). On the graph, the point where the two lines of the equation intersect is the only solution.
• The inconsistent system has no solution. On the graph, the lines never meet and are hence parallel.
• In the dependent system, we have an endless number of solutions. On the graph, the lines are coincident, which means that it is the same line, therefore any coordinate pair along it is a solution to both equations.

Solving a System of Equations

There are three main ways by which a system of linear equations can be solved which are the Elimination method (or addition method), the Substitution method and the Graphical method.

Substitution Method: The substitution method for solving systems of equations is a technique for simplifying a system of equations by expressing one variable in terms of another, removing one variable from the equation. The equation becomes solved when the simplified equation has just one variable to work with. The phases in the substitution method are as follows:

1. Solve for one of the variables in terms of the others in the first equation.
2. Fill in the blanks in the remaining equations with this expression.
3. Continue until the system has been reduced to a single linear equation.
4. Solve this problem and then back-substitute it till you get the solution.

Elimination Method: In a system of equations, the elimination method is used to eliminate a variable so that the remaining variable(s) may be solved. The steps of the elimination process are as follows:

1. Rewrite the equations so that the variables are in the correct order.
2. Modify one equation so that when the equations are combined together, the variable in both equations cancels out.
3. Combine the equations and remove the variable.
4. Calculate the value of the last variable.
5. Back-substitute the other variable and solve.

Graphical Method: The steps involved in the graphical method are as follows:

1. Create a graph for the first equation.
2. On the same rectangular coordinate system, graph the second equation.
3. Determine whether the lines are parallel, intersect, or are the same.
4. Determine the system’s solution.
5. Identify the intersection point if the lines cross. This is the system’s solution.
6. The system has no solution if the lines are parallel.
7. The system has an endless number of solutions if the lines are identical.
8. In both equations, double-check the solution.

Solved Examples Based on the Above Methods

Example1 (using substitution)

Solve the following equations using the substitution. 

-x+y=-5

2x-5y=1

We first solve for y

-x+y=-5

 y=x-5

Now, we substitute the value that we found in the terms of x in the second equation.

2x-5y=1

2x-5(x-5)=1

2x-5x+25=1

-3x= -24

 x= 8

Then we substitute x=8 into the first equation and solve for y.

-8+y=-5

y= 3

Therefore the solution is (8,3).

Example 2 (using the Elimination method)

Solve the following set of equations by the elimination method.

4x+y=8

2y+x=9

We line up the variables so that equations can be added easily in the later step.

4x+y=8

x+2y=9

We set up the variables in such a way that adding them will cancel out the system. If the variables do not match we multiply one equation by a scalar, to cancel out, and eliminate one of the variables. Here, we multiply the above equation by -2 and the y variable can be eliminated, then we add the equations. 

-2(4x+y=8)

x+2y=9

Result:

-8x-2y=-16

x+2y=9

Now we add the equations and eliminate the variable y.

 -8x-2y-2y =-16 +9

-7x=-7

Finally, solve for the variable x

-7x=-7

x=-7-7

x=1

Now, we substitute the value for x in any of the given equations. 

4x+y=8

4(1)+y=8

4+y=8

y=4

So, we get the solution of the equation (1,4)

Applications of System of Linear Equations

Many real-life situations with several constraints on the same variables can be solved using systems of equations.

• A system of equations, often known as simultaneous equations, is a collection of multiple-variable equations. A set of values that satisfy all of the equations in a system of equations is called an answer, and there can be many such solutions for any given system. The most common format for answers is in the form of an ordered pair: ( x, y). Substitution and elimination, as well as graphical approaches, are used to solve a system of equations. Systems of equations have a variety of practical uses. 
• When there are several constraints to consider, a system of equations can be utilized to solve a planning problem.
• It is also used to find the unknown quantities in any given system.
• It is also used in investigating the profits in stocks and trading.

Conclusion

A system of linear equations is made up of two or more equations with two or more variables, with all of the equations in the system being 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 many solutions, and incompatible systems have no solution. Real-world problems with multiple variables, such as those involving revenue, cost, and profit, can be solved using systems of equations.