A linear equation is an equation whose graph is a straight line. An equation of the form \(ax + by + c = 0\), where a, b, c are real numbers and a ≠ 0, b ≠ 0 is a general linear equation in two variables x and y. For example, 5x + 2y = 4, \(\frac{1}{2} x + 6y = 10\) are linear equations in x and y.
Follow the below steps to graph the linear equation in two variables:
1. Write an equation in form of showing one variable in terms of the other. For example, equation 5x + y = 14 can be written as y = 14 − 5x
2. Find at least three sets of values for these variables. In the above equation find a set of values for x and y.
| x | 1 | 2 | 3 |
| y | 9 | 4 | -1 |
Ordered pairs: (1, 9), (2, 4), (3, -1)
3. Draw the x and y-axis and define your scale to plot these three points in the graph.
4. Join these three points (1, 9), (2, 4), (3, -1)
5. You will get a straight line passing through them.
Graph of the first-degree equation in only one unknown quantity y = k is the line parallel to the x-axis at a distance of k units from it. Similarly, equation x = k is the line parallel to the y-axis at a distance of k units for it.
Example: Below graph represent x = 3 and y = 5. For equation x = 3, the value of x is 3 for any value of y, similarly for equation y = 5, the value of y is 5 for any value of x.
Set of equations with two or more variables in which the number of equations is the same as the number of variables is called a system of equations. Equations that have more than one unknown can have an infinite number of solutions. For example, x + y = 20 can be true for many pairs of x and y. Like (1) x =10, y = 10 (2) x = 12, y = 8 (3) x = 13, y = 7 etc.
If another equation is used alongside it, it is possible to find the only pair of values that solve both equations at the same time. These are known as simultaneous equations. In other words:
Two equations whose graphs intersect at a point named by an ordered pair of numbers that satisfies both equations are called simultaneous equations.
The coordinates of the point of intersection give the common solution of the two given linear equations. Let us see how to find the unknown variable values graphically using two linear equations.
Example: Solve graphically 2x − y = 6, x + y = 12
| x | 1 | 2 | 4 | 7 |
| y | -4 | -2 | 2 | 8 |
| x | 1 | 2 | 0 | 7 |
| y | 11 | 10 | 12 | 5 |
Plot these points and join them to get a straight line representing the equation.
Read the coordinates of the point of intersection. Here they are (6,6), therefore x = 6, y = 6 solves both equations.
The distance d between the point P with coordinates (x1,y1) and Q with coordinates (x2,y2) is
\(d = \sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2}\)
Therefore the distance of the point P from the origin is \(d = \sqrt{(x_1 - 0)^2 + (y_1 - 0)^2} = \sqrt{x_1^2 + y_1^2}\)
Example: Find the distance between points (7, 9), (4, 5)
\(d = \sqrt{(7-4)^2 + (9 -5)^2} \\ d = \sqrt{9 + 16} \\ d = \sqrt25 = 5\)
Answer: The distance between two points is 5 units.
Example: The coordinates of the vertices of a side of a square are (1, 2) and (3, 8). What is its area?
The length of the side is
\(S = \sqrt{(1-3)^2 + (2 -8)^2} \\ S = \sqrt{4 + 36} = \sqrt{40} \)
The area of a square is S2 = \({(\sqrt{40})}^2\)
Answer: The area of the square is 40 square units.
