4.2.1. Examples of Linear Equations in Two Variables

In a One-day International Cricket match between India and Sri Lanka played in Nagpur, two Indian batsmen together scored 176 runs. Express this information in the form of an equation. Here, you can see that the score of neither of them is known, i.e., there are two unknown quantities. Let us use x and y to denote them. So, the number of runs scored by one of the batsmen is x, and the number of runs scored by the other is y. We know that x + y = 176, which is the required equation.

In this example, we translate a real-world situation into a mathematical equation. Two players have scores that add up to a total (176 runs). We denote unknown scores with variables x and y. By defining the total equation, we articulate the relationship between both players' scores using an algebraic expression, leading to the equation x + y = 176.

Think of it like sharing a pizza between two friends. If you know that together they ordered a pizza with a total of 8 slices, but don’t know how many slices each person took, you could let x represent the slices taken by one friend and y the slices taken by the other. Thus, you can express the idea as x + y = 8.

This is an example of a linear equation in two variables. It is customary to denote the variables in such equations by x and y, but other letters may also be used. Some examples of linear equations in two variables are: 1.2s + 3t = 5, p + 4q = 7, πu + 5v = 9 and 3 = 2x – 7y. Note that you can put these equations in the form 1.2s + 3t – 5 = 0, p + 4q – 7 = 0, πu + 5v – 9 = 0 and 2x – 7y – 3 = 0, respectively.

A linear equation in two variables is an equation that can be expressed in the form ax + by + c = 0. Here, 'a', 'b', and 'c' are constants, and 'x' and 'y' are the variables. This format helps in identifying relationship patterns. For example, components like 1.2s + 3t = 5 can be rearranged into canonical form, which is useful for further analysis and graphing.

Imagine you are budgeting your monthly purchases. Let x represent the money spent on food and y represent the money spent on entertainment. Your budget can be written as an equation like 50x + 30y = 1000. Rearranging it into the standard form helps you understand how much you can allocate towards each aspect.

So, any equation which can be put in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero, is called a linear equation in two variables. This means that you can think of many many such equations.

For an equation to be classified as a linear equation in two variables, it must meet specific criteria: it should be expressible in the standard form provided, and at least one of the coefficients a or b must be non-zero. This emphasizes that a linear equation sketches out straight-line relationships which can either intersect, be parallel, or meet at a common point in geometric representations.

Consider designing a simple mobile plan where the costs depend on the minutes used (x) and the texts sent (y). If your plan states that each minute costs $2 and each text costs$1, your total expense can be expressed with an equation like 2x + 1y = total_cost, demonstrating how different usages impact overall spending.

Example 1 : Write each of the following equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case: (i) 2x + 3y = 4.37 (ii) x – 4 = 3y (iii) 4 = 5 x – 3y (iv) 2 x = y

In this example, the task is to rearrange several equations to match the standard form ax + by + c = 0 and identify coefficients a, b, and c. By manipulating the equations through addition, subtraction, or rearrangement, students learn how to express equations in a consistent format that facilitates solving and graphing.

Think of it like reorganizing a messy closet. You can’t find anything until you sort it out. By putting everything in its specific place, or in this case, standard format, it becomes much easier to see what you have and make decisions on what to do next.

Equations of the type ax + b = 0 are also examples of linear equations in two variables because they can be expressed as ax + 0. y + b = 0.

Equations that can be reformulated into the standard form with one variable having a coefficient of zero are still considered linear equations in two variables. This highlights the flexibility of interpreting linear relationships, even when one variable is absent, as the equation still represents a straight line when graphed.

Imagine a scenario where you’re tracking your savings. If you know your savings goal (like $100 by a certain date), you can express your savings progress as an equation: 200 - y = 0, where y represents how much you've saved. Even though this equation seems to deal with a single variable in appearance, it effectively describes a relationship involving a secondary component (time, in this case).

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