Solutions of Linear Equation refer to the set of values of the variables in the linear equations giving all possible solutions. Linear equations involve unknown quantities in the form of one or more variables to represent real-life problems. It helps in finding out the cost, mileage, speed, and distance, etc. with ease. We all use linear equations in our daily life without knowing. Show
In this lesson, let us learn in detail about the solutions of linear equations, types of solutions, and how to find the solutions, etc. What are the Solutions of a Linear Equation?The solutions of linear equations are the points at which the lines or planes representing the linear equations intersect or meet each other. A solution set of a system of linear equations is the set of values to the variables of all possible solutions. For example, while solving linear equations one can visualize the solution of a system of simultaneous linear equations by drawing 2 linear graphs and finding out their intersection point. The red line represents all the solutions for equation 1, and the blue line, solutions for equation 2. The intersection at a unique point (2,4) is the solution that will satisfy both equations. Types of Solutions for Linear EquationsA system of linear equations can have 3 types of solutions. Unique Solution of a system of linear equationsThe unique solution of a linear equation means that there exists only one point, on substituting which, L.H.S and R.H.S of an equation become equal. The linear equation in one variable has always a unique solution. For example, 3m =6 has a unique solution m = 2 for which L.H.S = R.H.S. Similarly, for simultaneous linear equations in two variables, the unique solution is an ordered pair (x,y) which will satisfy both the equations. No SolutionA system of linear equations has no solution when there exists no point where lines intersect each other or the graphs of linear equations are parallel. Infinite Many SolutionsA system of linear equations has infinitely many solutions when there exists a solution set of infinite points for which L.H.S and R.H.S of an equation become equal, or in the graph straight lines overlap each other. How to find the Solution of a Linear Equation?Solutions for Linear Equations in One VariableConsider the equation, 2x + 4 = 8
Hence, the solution of the equation 2x + 4 = 8 is x=2. Solutions for Linear Equations in Two VariablesThe following methods can be used to find the solutions of linear equations of two variables. Substitution Method Consider the following pair of linear equations, let's solve the following linear equations. x + y = 4 and x - y = 2
It should be clear why this process is called substitution. We express one variable in terms of another using one of the pair of equations and substitute that expression into the second equation. Elimination Method Consider the following pair of linear equations: 2x + 3y - 11 = 0, 3x + 2y - 9 = 0 The coefficients of x in the two equations are 2 and 3 respectively. Let us multiply the first equation by 3 and the second equation by 2, so that the coefficients of x in the two equations become equal:
Now, let us subtract the two equations, which means that we subtract the left-hand sides of the two equations, and the right-hand side of the two equations and the equality will still be preserved. 6x + 9y - 33 = 0 ,6x + 4y - 18 = 0 0 + 5y - 15 = 0, 5y = 15, y = 3 Note how x gets eliminated, and we are left with an equation in y alone. Once we have the value of y, we proceed as earlier – we plug this into any of the two equations. Let us put this into the first equation: 2x + 3y - 11 = 0, 2x + 3 (3) - 11 = 0, 2x + 9 - 11 = 0\, 2x = 2, x = 1 Thus, the non trivial solution is: x = 1, y = 3 Graphical Method As an example, lets solve the following linear equation: x - y + 2 = 0, 2x + y - 5 = 0. We draw the corresponding lines on the same axes: The point of intersection is (1,3), which means that x = 1, y = 3 is a solution to the pair of linear equations given by (2). In fact, it is the only solution to the pair, as two non-parallel lines cannot intersect at more than one point. Important Notes You can check directly about the types of solutions using the following conditions:
FAQs on Solutions of Linear EquationsHow To Solve the System of Linear Equations?We have different methods to solve the system of linear equations:
What Is a Unique Solution for a Linear Equation?The unique solution of a system of linear equations is an ordered pair or point which makes equality true in the equation. What Happens If a Pair of Linear Equations Is Consistent?If a pair of linear equations is consistent then the lines either intersect or coincide with (overlap) each other. What Are the 3 Solutions To Linear Equations?There are three ways to solve systems of linear equations: substitution, elimination, and graphing How Do You Find the Solution of a Linear System?
The lines intersect at zero points. How Do You Find the Solution of Two Linear Equations?Solving Systems of Equations by Substitution
How Do You Solve Linear Equations One Variable?
How Many Solutions of the Linear Equation 2x-5y=7 Are There?In the given equation 2x – 5y = 7, for every value of x, we get a corresponding value of y and vice-versa. Therefore, the linear equation has infinitely many solutions. How To Find Ordered Pair Solutions of Linear Equations?To figure out if an ordered pair is a solution to an equation, you could perform a test. Identify the x-value in the ordered pair and plug it into the equation. When you simplify, if the y-value you get is the same as the y-value in the ordered pair, then that ordered pair is indeed a solution to the equation. How Do You Find the Ordered Pair Solution of a Linear Equation in Two Variables?To check that an ordered pair is a solution, substitute the corresponding x- and y-values into each equation and then simplify to see if you obtain a true statement for both equations. How do you determine if there is one solution infinitely many solutions or no solution?We can identify which case it is by looking at our results. If we end up with the same term on both sides of the equal sign, such as 4 = 4 or 4x = 4x, then we have infinite solutions. If we end up with different numbers on either side of the equal sign, as in 4 = 5, then we have no solutions.
Is 0 0 infinite or no solution?Since 0 = 0 for any value of x, the system of equations has infinite solutions.
What is the formula of infinite solution?An infinite solution has both sides equal. For example, 6x + 2y - 8 = 12x +4y - 16. If you simplify the equation using an infinite solutions formula or method, you'll get both sides equal, hence, it is an infinite solution.
What is the formula for no solution?If (a1/a2) = (b1/b2) ≠ (c1/c2), then there will be no solution. This type of system of equations is called an inconsistent pair of linear equations. If we plot the graph, the lines will be parallel and system of equations have no solution.
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