One lock with many keys? Infinite solutions to one equation!

by ddos · Published · Updated

Algebra has been an integral part of our curriculum since your early school days. Algebra has taught us that it’s possible to find any unknown value in mathematics with certain conditions. These conditions that we call expressions.

Often represented with alphabets such as x and y, for example. Now, it’s not always the case that we get a definitive answer for every expression. Primarily, when we talk about finding solutions from any given expression, there are always three types of solutions that you can expect.

  1. A unique solution
  2. An infinite solution
  3. No solution

Now, while we’ll be discussing Infinite Solutions Maths in this blog, let’s quickly run through the other two types of answers as well in a nutshell for a quick revision. The first and the most common type of answer is a definitive answer.

Any expression that has given a fixed value in any given circumstance is regarded as unique solution. Whereas in the exact opposite nature, if any expression provides you with no definite/unique value in any regard, it is termed as no solution math.

You must be wondering if there is any other way to relate the types of solutions. The answer to this is yes! Algebra also moulds your geometry! As you already know, every line that you draw in any figure can be expressed in terms of x-intercept and y-intercept.

  • Here intercepts are the coordinates of the x-axis and y-axis in the graph. And as you might have guessed by now, each of your expressions dealing x and y with an ‘=’ can be drawn as a line on a graph.
  • Now, while you are solving any equation, let’s say 4x 5y = 7. Here, since you have two variables, you’ll require two expressions/equations to solve the value of x and y.
  • Also, as you might have learned in your initial geometry lesions, that any line can be expressed in the form of y = mx c.
  • Here, ‘y’ is called the slope of the line or graph; x stands for the x-intercept or coordinate on the x-axis, whereas m and c are constants depending on the graph.

You might ask, what’s the significance of it? When you talk about different types of solutions, you are indirectly comparing the y-intercept of the two expressions! Isn’t that interesting!

When does an expression comprise infinite solutions?

Since we’ll be discussing the infinite solutions in this topic, let’s go through the definition once. Any set of expressions where you can solve the equation with any value for the variables is considered to have infinite solutions.

Let’s take an example of it for simplification.

Ex1:

  1. y = x + 4
  2. 4y = 4x + 16

Here, as you can see, a pattern between the two expressions. If you take ‘2’ as a common multiplier from the second equation, you essentially end up with the first equation and vice versa. This is a simplified version of the equation to give you an idea of what infinite solutions might look like.

Now let’s see why a particular set of equations give infinite answers, and others do not. In very simplified terms, if you explain, you get two sets of lines on the graph whenever you have two sets of equations. The number of intersections these two lines have is the number of solutions for the two concerned equations.

In the case of straight lines, they can either intersect once, or have no intersection at all, i.e. parallel lines, or they both are superimposed on each other. The point of intersection, i.e. the coordinates of x and y of intersection, is the values of variables that will solve the equation.

If the lines are superimposed on each other, their y slope is the same in the y = mx c formula. And if they have the same y-intercept and slope, then no matter the value you use for the equation, you’ll be able to solve the equation, hence, infinite solutions.

So in terms of graphs, you can also state the definition of infinite solutions as two lines that superimpose on each other when drafted in algebraic expression will give infinite solutions.

Now let’s take some examples to understand the theory that you have read so far.

Let’s start with the standard format of any linear equation:

a1x + b1y +c1 = 0

a2x +b2y + c2 = 0

Here, a1, a2, b1, b2, c1 and c2 are numeric values.

For example:

Ex2:

3x – 5y – 20 = 0

6x – 10y – 40 = 0

Now, to satisfy the criteria of an infinite solution, the following case should be met.

(a1/a2) = (b1/b2) = (c1/c2)

i.e. all three ratios should be equal. Let’s take ex2, for example. Once we replace the values of a1, a2, b1, b2, c1 and c2, we get:

(3/6) ; (5/10) ; (20/40)

Which in turn gives:

(1/2) ; (1/2) ; (1/2)

Which, are equal, hence satisfying the criteria of (a1/a2) = (b1/b2) = (c1/c2)

Some examples:

Let’s solve this equation through arithmetic means to understand better.

=> 3x – 5y – 20 = 0

=> 5y = 3x – 20

=> y = (3/5)x – 4 —–> (Let’s take this as equation 1)

=> 6x – 10y – 40 = 0

=> 10y = 6x – 40

=> y = (3/5)x – 4 —–> (Let’s take this as equation 2)

As you can see, for both the equations, we get the same value of y, hence the equation has infinite solutions.

Let’s take another example with a single variable:

=> 3x 6x 3 = 3 (3x 1)

=> 9x 3 = 9x 3

=> 9x = 9x

Since both the right and left sides of the equation are the same, this equation will have infinite solutions; no matter what value of x you take, the equation will always be balanced.

One fun fact: 

Now that we have seen two linear equations being compared, what about one single equation?

What would be the case of ay = bx c ; (where a, b and c are real numbers).

Well, in all such cases, or you can say, all linear equations in two variables have infinite solutions. You might ask why? The graph of any linear equation in two variables where a and b are not both zero is a straight line. And as we have read earlier here, a straight line has an infinite number of solutions.

Hope you have understood the concept of infinite solutions now properly!