System of Linear Equations - Algebra (2024)

Linear equations are fundamental building blocks in mathematics, and a system of linear equations involves multiple such equations working together.

This article provides an overview of the system of linear equations and introduces the elimination and substitution methods as techniques to solve them.

System of Linear Equations - Definition

A system of linear equations is a collection of two or more linear equations that involve the same variables. In simple terms, we are trying to find the values of the variables that satisfy all of the equations simultaneously.

A system of linear equations can have one unique solution, infinitely many solutions, or no solution at all.

Graphical Representation

Before diving into the solution methods, it is helpful to visualize a system of linear equations through graphical representation. Each linear equation can be represented as a straight line on a Cartesian plane.

The point(s) at which the lines intersect represents the solution to the system of linear equations. If the lines never intersect, there is no solution. If the lines coincide, there are infinitely many solutions.

System of Linear Equations - Algebra (1)

Methods to solve a system of linear equations:

There are quite a number of methods to solve linear equations. For example:

  1. Substitution method
  2. Elimination method
  3. Cramer’s rule
  4. Matrix method
  5. Row reduction
  6. Iterative methods e.t.c

But the easiest and most common among the math folk are the first two. Let’s learn more about elimination and substitution methods.

Elimination Method

The process of solving a system of linear equations involves simplifying a pair of equations containing two variables into a single equation with one variable. The elimination method helps us achieve this goal by either adding or subtracting the equations, thus eliminating one variable in the process.

Consider this distinct system of linear equations:

4x - 2y = 10

3x + 2y = 14

Using the elimination method, we ensure that the equality of the equations is maintained while performing operations on both sides.

In this example, the two equations can be directly added because the coefficients of the y variable in both equations are additive inverses (+2 and -2). By adding the two equations, the y variable is removed:

(4x - 2y) + (3x + 2y) = 10 + 14

7x = 24

Now we have a simplified equation with only one variable, x, which can be solved:

7x = 24

x = 24/7

With the value of x determined, substitute it back into either of the original equations to solve for y. Let's use equation 1:

4(24/7) - 2y = 10

96/7 - 2y = 10

Now we can solve for y:

2y = 96/7 - 70/7

y = (96 - 70) / (2 * 7)

y = 13/7

So, the solution to the system of linear equations is x = 24/7 and y = 13/7.

At times, you might encounter a system of linear equations where the coefficients cannot be directly eliminated. In these cases, you'll need to manipulate one or both equations by either multiplying or dividing to match a pair of coefficients.

How to use the elimination method?

To apply the elimination method for solving a system of linear equations, follow these steps:

  1. Find a pair of terms in the system with the same variable and coefficients of equal magnitude. If necessary, rewrite one or both equations to match the coefficients.
  2. Add or subtract the two equations in the system to eliminate the terms identified in Step 1, resulting in a linear equation with only one variable.
  3. Solve the linear equation to obtain the variable's value.
  4. Insert the value of the determined variable back into one of the original equations to find the value of the remaining variable.

Substitution Method

The substitution method is another approach to solving a system of linear equations that involves isolating one variable in one equation and substituting its expression into the other equation. This method simplifies the problem into a single equation with one variable, making it easier to solve.

Let's consider a different system of linear equations:

x + y = 9

2x - 3y = 4

In this example, start by isolating one variable in one of the equations, say x in equation 1:

x = 9 - y

Now that the expression for x is in terms of y, substitute this expression into the other equation (equation 2) in place of x:

2(9 - y) - 3y = 4

18 - 2y - 3y = 4

This substitution transforms the equation into a single-variable linear equation, which can now be solved:

18 - 5y = 4

5y = 14

y = 14/5

With the value of y determined, substitute it back into the expression we found for x:

x = 9 - 14/5

x = (45 - 14) / 5

x = 31/5

So, the solution to the system of linear equations is x = 31/5 and y = 14/5.

The substitution method is especially useful when one variable can be easily isolated in one of the equations or when one of the variables already has a coefficient of 1 or -1.

How to use the substitution method?

To use the substitution method for solving a system of linear equations, follow these steps:

  1. Isolate one variable in one of the equations (express the variable in terms of the other variable).
  2. Substitute the isolated variable's expression into the other equation in the system, resulting in a single-variable linear equation.
  3. Solve the single-variable linear equation to determine the value of the variable.
  4. Replace the value of the found variable back into the expression for the isolated variable in the first equation to find the value of the other variable.

FAQs:

Q.1 When should I use the elimination method, and when should I use the substitution method?

The choice between elimination and substitution methods is often based on personal preference or the specific characteristics of the given system.

The elimination method might be more straightforward when the coefficients of one variable are equal or additive inverses, while the substitution method can be useful when one equation can be easily solved for one variable.

Q.2 How do I know if a system of linear equations has no solution or infinitely many solutions?

If you manipulate the equations and end up with a contradiction, such as 0 = 1, the system has no solution. If you obtain an identity, like 0 = 0, after manipulation, the system has infinitely many solutions.

Q.3 Can these methods be applied to systems with more than two variables?

Yes, both the elimination and substitution methods can be extended to solve systems with more than two variables. However, the process may be more complex and involve additional steps.

System of Linear Equations - Algebra (2024)

FAQs

What are the possible answers for a system of linear equations? ›

There are three types of systems of linear equations in two variables and three types of solutions.
  • An independent system has exactly one solution pair (x,y) . The point where the two lines intersect is the only solution.
  • An inconsistent system has no solution. ...
  • A dependent system has infinitely many solutions.

How many solutions does the system of linear equations have? ›

A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line).

How to determine the number of solutions in a linear system linear algebra? ›

A linear equation could have exactly 1, 0, or infinite solutions. If we can solve the equation and get something like x=b where b is a specific number, then we have one solution. If we end up with a statement that's always false, like 3=5, then there's no solution.

What's the easiest way to solve systems of linear equations? ›

SOLVE A SYSTEM OF LINEAR EQUATIONS BY GRAPHING.
  1. Graph the first equation.
  2. Graph the second equation on the same rectangular coordinate system.
  3. Determine whether the lines intersect, are parallel, or are the same line.
  4. Identify the solution to the system. ...
  5. Check the solution in both equations.
Nov 25, 2022

What is not a possible solution to a system of linear equations? ›

No solution: This occurs when the two lines are parallel and do not intersect. In this case, there is no common solution to the system. Infinitely many solutions: This happens when the two lines are coincident, meaning they are the same line.

What are the possible solutions to the system of equations? ›

The three possible solutions to a system of equations are one solution, infinite solutions, or no solutions. One solution means a single point satisfies the system. Infinite solutions mean an infinite number of points satisfy the system. No solution means that no points satisfy the system.

How to get infinite solutions in a system of equations? ›

An 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.

How to find out how many solutions a system of equations has? ›

A system of two equations can be classified as follows: If the slopes are the same but the y-intercepts are different, the system has no solution. If the slopes are different, the system has one solution. If the slopes are the same and the y-intercepts are the same, the system has infinitely many solutions.

How many equations should a system of linear equations have? ›

In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables.

How do you find all the solutions to a linear equation? ›

The steps for solving linear equations are:
  1. Simplify both sides of the equation and combine all same-side like terms.
  2. Combine opposite-side like terms to obtain the variable term on one side of the equal sign and the constant term on the other.
  3. Divide or multiply as needed to isolate the variable.
  4. Check the answer.
Oct 6, 2021

How many solutions can you find in a linear equation in one variable? ›

Linear Equation in One Variable Definition

It is of the form ax + b = 0, where x is the variable. This equation has only one solution.

How many solutions are there for each of the linear equations in two variables? ›

A linear equation in two variables will have infinite solutions.

What is the trick to solving linear equations? ›

To solve linear equations, find the value of the variable that makes the equation true. Use the inverse of the number that multiplies the variable, and multiply or divide both sides by it. Simplify the result to get the variable value. Check your answer by plugging it back into the equation.

What is the algebraic method for solving systems of linear equations? ›

These methods for finding the solution of linear equations are:
  1. Graphical Method.
  2. Elimination Method.
  3. Substitution Method.
  4. Cross Multiplication Method.
  5. Matrix Method.
  6. Determinants Method.
Apr 6, 2020

What is the best method to solve the linear system? ›

The substitution method: If you prefer to work things out using equations, the substitution method may be more suitable. We can start by solving one of the two linear equations for y in terms of x. Next, we can substitute that expression for y in the second linear equation. This will leave us with an equation in x.

What can a system of linear equations be solved by? ›

The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows: In the first equation, solve for one of the variables in terms of the others. Substitute this expression into the remaining equations.

What are the possibilities of a solution of a system of linear equations? ›

Types of Solutions for Linear Equation. Unique Solution. No Solution. Infinitely Many Solutions.

What are the basic solutions of system of linear equations? ›

basic solution: For a system of linear equations Ax = b with n variables and m ≤ n constraints, set n − m non-basic variables equal to zero and solve the remaining m basic variables. basic feasible solutions (BFS): a basic solution that is feasible. That is Ax = b, x ≥ 0 and x is a basic solution.

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