Question

How to study pair of linear equation in two variable

Answer 1

’s look at the solutions of some linear equations in two variables. Consider the equation 2x + 3y = 5. There are two variables in this equation, x and y.

Scenario 1: Let’s substitute x = 1 and y = 1 in the Left Hand Side (LHS) of the equation. Hence, 2(1) + 3(1) = 2 + 3 = 5 = RHS (Right Hand Side). Hence, we can conclude that x = 1 and y = 1 is a solution of the equation 2x + 3y = 5. Therefore, x = 1 and y = 1 is a solution of the equation 2x + 3y = 5.Scenario 2: Let’s substitute x = 1 and y = 7 in the LHS of the equation. Hence, 2(1) + 3(7) = 2 + 21 = 23 ≠ RHS. Therefore, x = 1 and y = 7 is not a solution of the equation 2x + 3y = 5.

Geometrically, this means that the point (1, 1) lies on the line representing the equation 2x + 3y = 5. Also, the point (1, 7) does not lie on this line. In simple words, every solution of the equation is a point on the line representing it.

To generalize, each solution (x, y) of a linear equation in two variables, ax + by + c = 0, corresponds to a point on the line representing the equation, and vice versa.

Pair of Linear Equations in Two Variables

Here is a situation: The number of times Ram eats a mango is half the number of rides he eats an apple. He goes to the market and spends Rs. 20. If one mango costs Rs.3 and one apple costs Rs.4, then how many mangoes and apples did Ram eat?

Let’s say that the number of apples that Ram ate is y and the number of mangoes is x. Now, the situation can be represented as follows:

y = (½)x … {since he ate mangoes (x) which were half the number of apples (y)}
3x + 4y = 20 … {since each apple (y) costs Rs.4 and mango (x) costs Rs.3}

Both these equation together represent the information about the situation. Also, these two linear equations are in the same variables, x and y. These are known as a ‘Pair of Linear Equations in Two Variables’.

To generalize them, a pair of linear equations in two variables x and y is:

a1x + b1 y + c1 = 0 and a2x + b2 y + c2 = 0.

Where a1, b1, c1, a2, b2, c2 are all real numbers and a12+ b12 ≠ 0, a22+ b22 ≠ 0. Some examples of a pair of linear equations in two variables are:

2x + 3y – 7 = 0 and 9x – 2y + 8 = 05x = y and –7x + 2y + 3 = 0

Answer 2

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✔️Linear equations in two variablesare equations which can be expressed as ax + by + c = 0, where a, b and c are real numbers and both a, and b are not zero. The solution of such equations is a pairof values for x and y which makes both sides of the equation equal.

✔️To solve a system of equations by graphing means to obtain the point of intersection (if any) of the graphs of each of the equation that make up the system. To graph a linear equation, we set the equation in the slope-intercept form and then graph the intercept and obtain the line using the slope

✔️Systems of linear equations can only have 0, 1, or an infinite number of solutions. These two lines cannot intersect twice. The correct answer is that the system has one solution.

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