While graphing two or more linear equations together, we will obtain a system of linear equations. It is a very important topic in mathematics. It plays a key role in algebra. So, it is essential to discuss this topic. The learning objectives of this topic are determining if an ordered pair is a solution of equations systems.
Second, we will talk about how to solve a system of equations by graphing. Apart from that, a complete explanation is provided with which one can define the number of solutions present in a linear system. Finally, we will be learning how to use these equations in different domains.
Determining if an ordered pair is a solution of equations systems
After understanding how to solve linear equations and inequalities, we should learn about the ways to solve linear equations using one variable. We should know that while substituting into an equation, its solution makes a true statement. It is because of the presence of values of variables.
Systems of linear equations
To form a system of linear equations, we should group two or more linear equations altogether. Now let us see systems of two linear equations in two unknowns. After doing so, we may solve equations containing larger systems. Given below is an example of a system of two linear equations.
In that, we use a brace at the start. It is to convey that both equations are grouped. It will form a system of equations.
4 x + y = 8 x – 3 y = 9
Infinite numbers of solutions are present in a linear equation in two variables. For example, 4 x + y = 8 contains solutions of infinite numbers. While plotting on the graph, it comes as a line. We should keep in mind that all the points present on the line act as a solution to the equations. Also, solutions to every equation are a point on the line.
If we need to solve the system of two linear equations, we have to identify the variables’ values. They are present as solutions to both equations. In simple terms, there are ordered pairs (x, y). These pairs can make both equations true. These are referred to as the solutions to a system of equations.
Solutions of a system of equations
To make all the equations true, we need to use this system of equations. An ordered pair (x, y) can be used to represent these equations. We substitute values into each equation to determine whether the ordered pair acts as a solution to a system of two equations. Therefore, if it is true, it is the solution to the system.
Take a look at the system below:
3 x – y = 7 x – 2 y = 4
We need to find whether the ordered pair (2, – 1) is a solution.
Substituting x = 2 and y = – 1 in both equations:
3 x – y = 7
3 (2) + 1 = 7
7 = 7
Therefore, it is true.
x – 2 y = 4
2 – 2 (- 1) = 4
4 = 4
Therefore, it is also true.
Hence, the above-mentioned ordered pair is true for both equations. Therefore, it is proved to be a solution to the system.
Consider another example. Here, a different ordered pair is given:
Let us see whether the ordered pair (3, 2) is a solution.
On substituting x = 3 and y = 2, we obtain:
3 x – y = 7
3 (3) – 2 = 7
9 – 2 = 7
7 = 7
Hence, it is true.
x – 2 y = 4
2 – 2 (2) = 4
– 2 = 4
Hence, it is false.
Only one of the equations is true for the ordered pair (3, 2). The other one is false. Therefore, it is not considered a solution to both equations. Hence, it cannot be a solution to the system.
Solving a system of linear equations by graphing
In this topic, we will utilize three major techniques to solve a system of linear equations. The first technique is graphing.
When we plot the graph for a linear equation, it forms a line. Every point plotted on the graph is considered a solution to the equation. Therefore, we will draw two lines if it is a system of two equations. With the help of that, we can see that all the points are solutions to every equation. We can easily find the solution to the system by simply identifying what the lines have in common.
Most equations have one solution in one variable. However, some equations have no solutions. They are termed contradictions. Moreover, there are solutions for all the equations. They are called identities.
Let us see the steps involved in solving graphing. With that, it is easy to understand
1. Solve the systems using graphing:
x – 5 y = – 5 2 x + y = 8
having an ordered pair as (3, 2)
2. Solve the systems using graphing:
– x + 2 y = 5 3 x + y = 12
having an ordered pair as (2, 3)
Given below are the steps one could use in solving a system of linear equations using the graphing technique:
- Start graphing the first equation.
- Then, start graphing the second equation. It has to be on the same coordinate system.
- After doing so, now demonstrate if the lines can intersect.
- Also, check if they are parallel or present on the same line.
- Find the solution to the system.
- Identify the intersection point if the lines intersect. Verify if the solution is for both equations. It is the system’s solution.
- The system contains zero solutions if the lines are parallel.
- The system contains an infinite number of solutions if they have the same line.
Defining the no. of solutions of a linear system
There are other ways to determine the solutions without graphing. Let us see how. Consider the equation that provides parallel lines:
y = 12 x – 3 x – 2 y = 4
While having the lines in slope-intercept form, it gives:
y = 12 x – 3 and y = 12 x – 2
From the above equation, one can understand that for both equations, it is not possible to have a single ordered pair (x, y). The name for this type of system is the inconsistent system. Hence, it consists of zero solutions. If there is at least one solution present in the system of equations, then it is called a consistent system.
In addition to that, one can also classify the equations in a system of equations as dependent and independent. Two equations have their own set of solutions if they are independent equations. Parallel and intersecting lines are independent. In contrast, if in one equation, all the solutions are solutions to one another, then these equations are dependent. We obtain coincident lines while graphing dependent equations.
Applications of graphing the systems of equations
Let us see how we modified the strategy to solve systems of equations.
Below is a problem-solving technique that one needs to use to solve linear equation systems.
- Start by reading the problem. Ensure all the ideas and phrases are clearly understood.
- Then, identify what we need to do with it. Understand what we are looking for.
- Name the variables that we looked for. Then choose appropriate variables which will represent those quantities.
- Translate the whole thing into a system. This system should contain equations.
- With the help of appropriate methods, solve the system of equations.
- Verify the problem and check whether the answers are correct. Also, ensure that they make sense.
- Finally, answer the questions using full sentences.
After understanding how to solve the problems, let us see in which places it is used:
- A system of equations is used in solving mixture problems. One can solve them by writing a linear equation system that will represent a mixture problem. And then solve the system. Finally, interpret the results.
- The system of equations plays a major role in solving value problems. There are two steps to do it:
- Start by writing a linear equation system that represents a number problem.
- Afterward, define and apply the most appropriate method to solve the system.
- Another important usage of the system of equations is seen in solving cost and revenue problems. Follow the steps mentioned below:
- Start by specifying the variables in a cost or revenue linear equation system.
- Second, demonstrate and apply the most appropriate method to solve the system.
- The next step is to write down the system of inequalities. These inequalities should represent the profit region.
- Finally, start interpreting the systems’ solutions to a cost or revenue system of equations and inequalities.
To conclude, graphing systems of equations is a widely used process in algebra. We have seen that it is possible to obtain a system of equations while graphing two or more linear equations together. We have verified whether an ordered pair acts as a solution to a system of equations.
Furthermore, we have also seen some example problems where we solved systems of linear equations. Ultimately, we have also discussed the possible applications of graphing a linear equation.