What is the solution if the lines are the same?
In Mathematics, we come across equations and expressions. An equation is an expression with an equal sign used in between. An expression is made up of variables and constant terms conjoined together using algebraic operators. An algebraic equation can have one or more solutions. The solution of the equation or the values of variables in the equation must satisfy the equation. In this article, we are going to discuss the equations with infinite solutions, and the condition for the infinite solution with examples. Show What are Infinite Solutions?The number of solutions of an equation depends on the total number of variables contained in it. Thus, the system of the equation has two or more equations containing two or more variables. It can be any combination such as
Depending on the number of equations and variables, there are three types of solutions to an equation. They are
The term “infinite” represents limitless or unboundedness. It is denoted by the letter” ∞ “. To solve systems of an equation in two or three variables, first, we need to determine whether the equation is dependent, independent, consistent, or inconsistent. If a pair of the linear equations have unique or infinite solutions, then the system of equation is said to be a consistent pair of linear equations. Thus, suppose we have two equations in two variables as follows: a1x + b1y = c1 ——- (1) a2x + b2y = c2 ——- (2) The given equations are consistent and dependent and have infinitely many solutions, if and only if, (a1/a2) = (b1/b2) = (c1/c2) Conditions for Infinite SolutionAn equation can have infinitely many solutions when it should satisfy some conditions. The system of an equation has infinitely many solutions when the lines are coincident, and they have the same y-intercept. If the two lines have the same y-intercept and the slope, they are actually in the same exact line. In other words, when the two lines are the same line, then the system should have infinite solutions. It means that if the system of equations has an infinite number of solution, then the system is said to be consistent. As an example, consider the following two lines.
These two lines are exactly the same line. If you multiply line 1 by 5, you get the line 2. Otherwise, if you divide the line 2 by 5, you get line 1. Infinite Solutions ExampleExample: Show that the following system of equation has infinite solution: 2x + 5y = 10 and 10x + 25y = 50 Solution: Given system of the equations is 2x + 5y = 10 and 10x + 25y = 50 2x + 5y = 10 ………….(1) 10x + 25y = 50 ………..(2) By comparing with linear system, we get a1x + b1y = c1 a2x + b2y = c2 => a1 = 2, b1 = 5, c1 = 10, a2 = 10, b2 = 25 and c2 = 50 Now, the ratios are: (a1/a2) = 2/10 = 1 / 5 (b1/b2) = 5 /25 = 1/5 (c1/c2) = 10/50 = 1/5 (a1/a2) = (b1/b2) = (c1/c2) Therefore, the given system of equation has infinitely many solutions. Stay tuned with BYJU’S – The Learning App and download the app for more Maths-related articles and explore videos to learn with ease. Graphing Systems of Linear Equations Learning Objective(s) · Solve a system of linear equations by graphing. · Determine whether a system of linear equations is consistent or inconsistent. · Determine whether a system of linear equations is dependent or independent. · Determine whether an ordered pair is a solution of a system of equations. · Solve application problems by graphing a system of equations. Introduction Recall that a linear equation graphs as a line, which indicates that all of the points on the line are solutions to that linear equation. There are an infinite number of solutions. If you have a system of linear equations, the solution for the system is the value that makes all of the equations true. For two variables and two equations, this is the point where the two graphs intersect. The coordinates of this point will be the solution for the two variables in the two equations. Systems of Equations The solution for a system of equations is the value or values that are true for all equations in the system. The graphs of equations within a system can tell you how many solutions exist for that system. Look at the images below. Each shows two lines that make up a system of equations.
When the lines intersect, the point of intersection is the only point that the two graphs have in common. So the coordinates of that point are the solution for the two variables used in the equations. When the lines are parallel, there are no solutions, and sometimes the two equations will graph as the same line, in which case we have an infinite number of solutions. Some special terms are sometimes used to describe these kinds of systems. The following terms refer to how many solutions the system has. o When a system has one solution (the graphs of the equations intersect once), the system is a consistent system of linear equationsand the equations are independent. o When a system has no solution (the graphs of the equations don’t intersect at all), the system is an inconsistent system of linear equations and the equations are independent. o If the lines are the same (the graphs intersect at all points), the system is a consistent system of linear equations and the equations are dependent. That is, any solution of one equation must also be a solution of the other, so the equations depend on each other. The following terms refer to whether the system has any solutions at all. o The system is a consistent system of linear equations when it has solutions. o The system is an inconsistent system of linear equations when it has no solutions. We can summarize this as follows: o A system with one or more solutions is consistent. o A system with no solutions is inconsistent. o If the lines are different, the equations are independent linear equations. o If the lines are the same, the equations are dependent linear equations.
Advanced Question Which of the following represents dependent equations and consistent systems? A) B) C) D) Verifying a Solution From the graph above, you can see that there is one solution to the system y = x and x + 2y = 6. The solution appears to be (2, 2). However, you must verify an answer that you read from a graph to be sure that it’s not really (2.001, 2.001) or (1.9943, 1.9943). One way of verifying that the point does exist on both lines is to substitute the x- and y-values of the ordered pair into the equation of each line. If the substitution results in a true statement, then you have the correct solution!
Remember, that in order to be a solution to the system of equations, the value of the point must be a solution for both equations. Once you find one equation for which the point is false, you have determined that it is not a solution for the system. Which of the following statements is true for the system 2x – y = −3 and y = 4x – 1? A) (2, 7) is a solution of one equation but not the other, so it is a solution of the system B) (2, 7) is a solution of one equation but not the other, so it is not a solution of the system C) (2, 7) is a solution of both equations, so it is a solution of the system D) (2, 7) is not a solution of either equation, so it is not a solution to the system Graphing as a Solution Method You can solve a system graphically. However, it is important to remember that you must check the solution, as it might not be accurate.
Which point is the solution to the system x – y = −1 and 2x – y = −4? The system is graphed correctly below. A) (−1, 2) B) (−4, −3) C) (−3, −2) D) (−1, 1) Graphing a Real-World Context Graphing a system of equations for a real-world context can be valuable in visualizing the problem. Let’s look at a couple of examples.
Note that if the estimate had been incorrect, a new estimate could have been made. Regraphing to zoom in on the area where the lines cross would help make a better estimate. Paco and Lisel spent $30 going to the movies last night. Paco spent $8 more than Lisel. If P = the amount that Paco spent, and L = the amount that Lisel spent, which system of equations can you use to figure out how much each of them spent? A) P + L = 30 P + 8 = L B) P + L = 30 P = L + 8 C) P + 30 = L P − 8 = L D) L + 30 = P L − 8 = P Summary A system of linear equations is two or more linear equations that have the same variables. You can graph the equations as a system to find out whether the system has no solutions (represented by parallel lines), one solution (represented by intersecting lines), or an infinite number of solutions (represented by two superimposed lines). While graphing systems of equations is a useful technique, relying on graphs to identify a specific point of intersection is not always an accurate way to find a precise solution for a system of equations. What is the solution if both lines are the same?If the two lines are the same, then every point on one line is also on the other line, so every point on the line is a solution to the system. The system has an infinite number of solutions, and the two equations are really just different forms of the same equation.
Is there a solution if the lines are parallel?If the lines are parallel, then the pair of equations has no solution.
How many solutions do two of the same lines have?If the graphs of the equations are the same, then there are an infinite number of solutions that are true for both equations. When the lines intersect, the point of intersection is the only point that the two graphs have in common.
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