System of Linear Equations Solutions

A system of linear equations is a set of two or more linear equations. They are typically made up of two or more variables such that all the equations can be considered simultaneously and can represent a single problem with two or more unknown variables. 

System of linear equations solutions refers to a set of values of these variables such that all the equations are satisfied. However, not all systems of linear equations may necessarily have a solution. Some may have unique solutions, while some may have an infinite number of solutions, and some may not have a solution at all. 

How can we identify the number of solutions? 

The number of solutions can be identified either graphically or algebraically. 

In graphical solutions, the solutions for the system of linear equations are simply the points on a graph where the lines of the two equations intersect. 

For this, the plotting of the equations on the graph is very necessary. 

How can we plot the graph of a system of linear equations to obtain their solutions? 

There are a few important concepts that need to be kept in mind before plotting a system of linear equations on a graph. They are: 

• The graphs are always plotted in the xy-plane, as a system of linear equations consists of two or more equations that have two variables. 

• The slope-intercept form of the equations, which is y=mx + b, tells us about the slope and intercept of the line. 

  • m is the slope of the line. If the slope is 0, the equation has only one variable.

  • b is the y-intercept.

The points at which these lines intersect on a graph tell us about the solutions to the system of linear equations. 

• If the lines intersect at a single point, there is a unique solution.

• If the lines intersect at multiple points, there are multiple solutions.

• If the lines are parallel, with the same slope and different y-intercepts, the lines will never intersect, and there are no solutions.

• If the lines have the same slope and same y-intercept, it can be assumed they are the same lines and hence will overlap in many places. Therefore, they have infinite solutions. 

How can we solve a system of linear equations algebraically?

Obtaining a system of linear equations’ solutions algebraically does not require graphical representation, but it does require the graphical equation form for the linear equations. 

The equations need to be converted into the y = mx + b format before proceeding with algebraic elimination or substitution.

1. Algebraic elimination

In algebraic elimination, the goal is to simply remove or eliminate one of the variables by subtraction or addition so that we can obtain the value of the other equation first.

For example, consider the two equations 2x+y=7 and x-y=2,

The algebraic elimination method would involve:

2x + y = 7

+ x – y = 2 

3x = 9

Therefore, x = 3

And, y = 1

2. Algebraic substitution

In algebraic substitution, the goal is to substitute one of the variables for the other so that the equations can be solved.

For example, consider the two equations, 3x + 4y = 18 and 2x – y = 1

First, we substitute one of the variables for another. Let’s consider variable y. 

So we get

2x – y = 1

y = 2x – 1

When we substitute this value of y in the other equation, we can solve it to obtain the value of variable x, which is as follows:

3x + 4 (2x-1) = 18

3x + 8x – 4 = 18

11x = 18 + 4

11x = 22

x = 2

Therefore, we obtain the value of x = 2

Now, simply we can enter the value of x in any of the equations to solve for the value of y. 

y = 2x-1

y = 2 (2) – 1

y = 4-1

y = 3 

Therefore, the value of y obtained is 3.

This way, we can obtain a system of linear equations’ solutions algebraically. 

How can we find solutions to a real-life problem using a system of linear equations?

We have learned how to solve equations of the system of linear equations graphically or algebraically. Typically, systems of linear equations hold importance because they are used to solve real-life problems. Similarly, let us solve a question that reflects a real-life example for better understanding.

Let’s consider the following problem:

A boy can run at a speed of 0.2 km per minute. A horse can run at 0.6 km per minute. However, the horse is allowed to start 6 minutes after the boy as it requires 6 minutes to saddle the horse. How long will the horse take to overtake the boy?

Let’s assume that the boy covers x distance in y time. 

As we know, distance = speed * time

For the boy, we get x = 0.2 y

Similarly, for the horse, we get x = 0.5 (y – 6)

Note: 6 is subtracted from the time by the horse as the horse starts 6 minutes later due to the time required to saddle it.

For solving this algebraically, let us use the substitution method. 

Since x = 0.2 y, and x = 0.5 (y – 6)

0.2y = 0.5 (y – 6)

0.2 y = 0.5 y – 3

0.3 y = 3

Therefore, we get y = 10 minutes.

The horse overtakes the boy within 10 minutes.

Conclusion

In this article, we have learned how to solve a system of linear equations using different methods. Systems of linear equations are an easy way of solving real-life problems, and hence, they hold great importance in physics. Apart from the solved examples, these also include problems of upstream and downstream velocities and majorly speed-related questions. For solving MCQs, preferably algebraic methods are used.