On Which Interval(S) Can Guarantee a Solution to the Equation F(X)=2 ?

Solutions of Linear Equation

Solutions of Linear Equation refer to the prepare of values of the variables in the linear equations giving all possible solutions. Linear equations involve unknown quantities in the form of i or more variables to represent real-life problems. It helps in finding out the price, mileage, speed, and altitude, etc. with ease. We all use linear equations in our daily life without knowing.

In this lesson, let usa acquire in detail about the solutions of linear equations, types of solutions, and how to find the solutions, etc.

ane.	What are the Solutions of a Linear Equation?
2.	Types of Solutions for Linear Equations
3.	How to Find the Solution of a Linear Equation?
iv.	Examples of Solutions of a Linear Equation
five.	FAQs on solutions of linear equation

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 i tin can visualize the solution of a system of simultaneous linear equations past drawing two linear graphs and finding out their intersection point.

The reddish line represents all the solutions for equation 1, and the blue line, solutions for equation two. The intersection at a unique signal (2,4) is the solution that will satisfy both equations.

Types of Solutions for Linear Equations

A system of linear equations tin can have 3 types of solutions.

Unique Solution of a organisation of linear equations

The unique solution of a linear equation means that there exists only i point, on substituting which, L.H.S and R.H.Due south of an equation go equal. The linear equation in one variable has e'er a unique solution. For example, 3m =half-dozen has a unique solution m = 2 for which L.H.S = R.H.Southward. Similarly, for simultaneous linear equations in two variables, the unique solution is an ordered pair (x,y) which volition satisfy both the equations.

No Solution

A system of linear equations has no solution when there exists no indicate where lines intersect each other or the graphs of linear equations are parallel.

Space Many Solutions

A system of linear equations has infinitely many solutions when in that location exists a solution ready of infinite points for which Fifty.H.S and R.H.S of an equation get equal, or in the graph direct lines overlap each other.

How to find the Solution of a Linear Equation?

Solutions for Linear Equations in I Variable

Consider the equation, 2x + 4 = 8

• To observe the value of x, start, we remove iv from L.H.S, so we decrease 4 from both sides of the equation. 2x + 4 - 4 = viii - 4
• Simply. Now we get, 2x = iv
• Now we have to remove 2 from Fifty.H.S in social club to become x, therefore nosotros divide the equation by 2. 2x/2 = 4/2, x=two

Hence, the solution of the equation 2x + 4 = 8 is 10=2.

Solutions for Linear Equations in Two Variables

The following methods can be used to find the solutions of linear equations of 2 variables.

Commutation Method

Consider the following pair of linear equations, let'south solve the following linear equations.

x + y = 4 and x - y = 2

• Permit's rearrange the outset equation to limited y in terms of x, as follows: 10 + y = 4, y = 4 - x
• This expression for y can now be substituted in the second equation, and so that we volition be left with an equation in ten alone: 10 - y = 2, x - 4 + x = 2, 2x = 6 x = half dozen/2, 10 = 3
• Once we accept the value of x, nosotros tin plug this dorsum into whatever of the two equations to find out y. Lets plug information technology into the first equation: x + y = 4 (3) + y = 4, y = 4 - 3 = 1, y = 1
• The last not fiddling solution is: 10 = three, y = 1

It should exist articulate why this process is called substitution. We express one variable in terms of some other using ane of the pair of equations and substitute that expression into the 2nd 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 2nd equation by 2, so that the coefficients of 10 in the two equations become equal:

• three {2x + 3y - 11 = 0} 6x + 9y - 33 = 0
• 2 {3x + 2y - nine = 0} 6x + 4y - 18 = 0

At present, let us subtract the two equations, which means that nosotros subtract the left-hand sides of the two equations, and the right-manus side of the two equations and the equality will yet be preserved.

6x + 9y - 33 = 0 ,6x + 4y - 18 = 0 0 + 5y - 15 = 0, 5y = 15, y = iii

Annotation how x gets eliminated, and nosotros are left with an equation in y alone. Once nosotros have the value of y, we keep as before – we plug this into any of the two equations. Let usa put this into the kickoff equation:

2x + 3y - 11 = 0, 2x + 3 (3) - xi = 0, 2x + nine - eleven = 0\, 2x = two, x = 1

Thus, the not niggling solution is: 10 = 1, y = iii

Graphical Method

Equally an example, lets solve the following linear equation: ten - y + two = 0, 2x + y - v = 0. Nosotros draw the corresponding lines on the aforementioned axes:

The point of intersection is (ane,3), which means that ten = 1, y = three is a solution to the pair of linear equations given by (ii). In fact, it is the only solution to the pair, as two non-parallel lines cannot intersect at more than one bespeak.

Of import Notes

You lot can check directly nigh the types of solutions using the post-obit weather:

• Unique solution (consistent and independent) a1/a2 ≠ b1/b2
• No solution (inconsistent and contained) a1/a2 = b1/b2 ≠ c1/c2
• Infinite many solutions (consistent and dependent) a1/a2 = b1/b2 = c1/c2

Examples of Solutions of Linear Equation

1. Example 1: Tom loves to collect ¢two and ¢5 coins in his piggy bank. He knows that the full sum in his piggy bank is ¢77, and it has iii times as many two-cent coins as five-cent coins in information technology. He wants to know the verbal number of ¢2 and ¢5 coins in his piggy banking concern. Tin can you help him find the count?

   Solution:

   Let the number of ¢5 coins be ten.
   The number of ¢2 coins will exist 3x.
   Total amount = 5x + 2(3x)
   Thus, 5x + 2(3x) = 77
   5x + 6x = 77
   11x = 77
   10 = 7

   Therefore, Tom has 7 coins ¢5 of and 21 coins of ¢two.

2. Example 2: In a river, a boat tin can travel twoscore miles upstream in 2 hours. The same gunkhole tin can travel 60 miles downstream in ii hours.

   • What is the speed of the gunkhole in all the same water?
   • What is the speed of the current?

   Solution:

   Let u.s.a. presume that:

   • the speed of gunkhole in notwithstanding water = 10 miles per hr
   • the speed of current = y miles per hr.

   During upstream, the current pulls back the boat's speed and the speed of the boat during upstream = \(ten-y\).

   During downstream, the current's speed adds to the boat's speed and the speed of the gunkhole during downstream = \(10+y\).

   Thus,

   	Distance(d)	Time(t)	Speed	Speed=d/t
   UPSTREAM	twoscore miles	two hr	ten - y	40/2=20
   DOWNSTREAM	60 miles	2 hour	x + y	60/2=thirty

   Using the last 2 columns of the table:

   x - y = twenty -------(1)

   x + y = 30 -------(ane)

   Adding (1) and (2):

   2x = 50, x =25

   Substitute this in (ii):

   y = xxx-25, y =5

   The speed of boat = 25 miles per 60 minutes
   The speed of current = v miles per hour

go to slidego to slide

Breakdown tough concepts through elementary visuals.

Math will no longer be a tough bailiwick, especially when you understand the concepts through visualizations.

Book a Free Trial Form

Practice Questions​​​​​​

go to slidego to slidego to slide

FAQs on Solutions of Linear Equations

How To Solve the Organisation of Linear Equations?

Nosotros have different methods to solve the system of linear equations:

• Graphical method
• Exchange method
• Cross-multiplication method
• Elimination method
• Determinants method

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 three Solutions To Linear Equations?

There are three ways to solve systems of linear equations: substitution, emptying, and graphing

How Do You Find the Solution of a Linear Organization?

• First, solve i linear equation for y in terms of x.
• Then substitute that expression for y in the other linear equation. You'll become an equation in x .

The lines intersect at nada points.
The lines intersect at exactly one signal.
The lines intersect at infinitely many points.

How Do You Find the Solution of Two Linear Equations?

Solving Systems of Equations by Substitution

• Solve one of the ii equations for i of the variables in terms of the other.
• Substitute the expression for this variable into the second equation, then solve for the remaining variable.

How Do Yous Solve Linear Equations One Variable?

• Pace i: Simplify each side, if needed.
• Step ii: Use Add./Sub. Properties to move the variable term to one side and all other terms to the other side.
• Step iii: Utilize Mult./Div. ...
• Step iv: Cheque your answer.

How Many Solutions of the Linear Equation 2x-5y=vii Are There?

In the given equation 2x – 5y = 7, for every value of x, we go 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, yous could perform a test. Place the x-value in the ordered pair and plug it into the equation. When y'all simplify, if the y-value y'all get is the same equally the y-value in the ordered pair, then that ordered pair is indeed a solution to the equation.

How Do Yous Find the Ordered Pair Solution of a Linear Equation in Two Variables?

To cheque that an ordered pair is a solution, substitute the corresponding x- and y-values into each equation and and then simplify to see if you lot obtain a true statement for both equations.

lyallrojectime.blogspot.com

Source: https://www.cuemath.com/algebra/solutions-of-a-linear-equation/

Share this post