A graph contains a grid network of x-axis and y-axis which is used to locate coordinate points. The value of a coordinate includes a pair in which the first value belongs to the X-axis and the second value belongs to the Y-axis. However, there are other ways of locating a coordinate point in a graph.
That being said, linear equations can also be used to locate coordinate points in a graph. But, what are linear equations? How can one solve them to locate coordinate points on a graph? Keep reading the headings below to find out the answers to various questions related to graphing a linear equation.
What is a graph?
Have you seen a graph? It looks like a plane divided into several sections made by the intersection of horizontal and vertical lines. In mathematics, a graph is defined as a pictorial representation of data or a diagram that indicates values in an organized structure. A graph is plotted only if both the values of the x-axis and the y-axis are known.
The point on a graph where the lines of these axes intersect is known as a coordinate point. This point is the result of solving linear equations.
A graph is usually drawn on a coordinate plane. It is important to know that a coordinate plane consists of grids made by the intersecting lines of the X-axis and Y-axis.
An x-axis and a y-axis also represent a graph. Along with this, a graph also has a number line or scale, which is used to locate the precise point in a coordinate grid.
One can take advantage of a graph to represent the data of various things. A graph can define the number of supplies, requirements, and other parameters used in day-to-day life. This way, it is easy to represent a given data in a systematic order.
What is a linear equation?
A linear equation is an equation that has 1 as the highest exponential degree. It means that all the variables present in a linear equation have an exponential value of 1.
Linear equations always form a straight line on a graph. It means a straight line can be drawn by solving a linear equation – along with one of the quadrants in a coordinate grid.
A linear equation is expressed as an algebraic equation. It means each component of the equation has 1 as its exponent. Apart from this, a linear equation always forms a straight line when graphed on a coordinate grid. A straight line is formed with a linear equation, whether it is a value on the x scale or the y scale.
Types of linear equations
There are two types of linear equations –
- Linear equations in one variable
- Linear equations in two variables
The formula for a linear equation
The formula for a linear equation is used to express a linear equation. However, this can also be done in various ways. It means one can define a linear equation in the standard form, point-slope form, and slope-intercept form.
However, if the standard form is used to express a linear equation, the results can vary based on the variables. It is also important to know that no matter what the form is, the exponential degree of all the variables present in a linear equation is always 1.
A linear equation in one variable has a standard form which is expressed by-
Ax + B = 0.
A = coefficient
x = variable
B = constant
Linear equation in two variables has a standard form which is expressed by-
Ax + By = C
A and B = coefficient
x and y = variables
C = constant
How to solve a linear equation?
To graph a linear equation on a coordinate plane, one should also know how to solve it. The solution comprises simple steps to locate a point on a graph.
An equation is different from other mathematical problems. It means that any creation has values on both sides to make it equal. It suggests that if we subtract a particular number from both sides of an equation, the equation remains the same and has the same value. The same goes for division, multiplication, and addition.
For example: 4x – 3 = 13
In this equation, one can solve it by moving constant values to one side and the variable values to the other side. By doing this, one can calculate the value of the unknown variable in the given equation.
4x – 3 = 13
4x = 13 + 3
4x = 16
x = 16/4
x = 4
From the above solution, it is clear that one should perform mathematical operations in such a way that the left-hand side is equal to the right-hand side. It is done so that the balance of the equation is not disturbed. To understand this balance, refer to the example below.
For example: in the linear equation 4x – 3 = 13, if the addition of +3 is made on both sides, then the resulting linear equation will be,
4x – 3 + 3 = 13 + 3
4x = 16
x = 16/4
x = 4
From the above calculation, we understand that even if there is an addition on both sides of an equation, the result does not change. It is a convenient way to graph a linear equation.
What do you understand by graphing a linear equation?
Since it is clear that a linear equation forms a straight line on the graph, this line can either be parallel to the x-axis or the y-axis.
To understand this, one can suppose if there is a linear equation with a variable that has a value from the x-axis, then to graph a linear equation; a straight line is drawn which is parallel to the y-axis.
In the same way, if a linear equation has a variable with a value from the y-axis, then a straight line is drawn, which is parallel to the x-axis.
How to plot a graph using a linear equation in one variable?
It is a simple task when it comes to plotting a graph using a linear equation. However, there are different methods of solving when the linear equation involves one variable and two variables.
Read the following method to understand how to solve a linear equation in one variable and plot it on a graph.
For the equation: 2x – 5 = 9
The solution is,
2x – 5 = 9
2x = 9 + 5
2x = 14
x = 14/2
x = 7
And since this is a positive value on the x-axis, a straight line can be drawn for the same.
How to plot a graph using a linear equation in two variables?
As the name suggests, these types of linear equations have two variables. Hence, the calculation differs from that of one variable.
Follow these steps to plot a graph for learning equations in two variables.
For the equation: x – 3y = 3, the steps will be –
- The first thing is to convert the given equation in y = mx + b form.
- It will give: y = x/3 – 1
- After getting this, replace the value of acts with other numbers. for example, 0, 1, 2…
- By substituting these values for x in the equation: y = x/3 – 1, one gets different values for y to create coordinates.
- When 0 is taken as a value for x, then the resulting equation will be, y = -1
- Likewise, if the value for x is taken as 3, the resulting equation will be, y = 0.
- In the same way, if the value of x is substituted as 6, the resulting linear equation will be, y = 1.
As these resulting equations satisfy the original equation of y = x/3 – 1, a graph can be plotted against it.
Some characteristics of linear equations in one variable.
- The variable of a linear equation and its value is called the root of the linear equation.
- The variable of a linear equation is also called the solution
- The resulting solution of a linear equation remains the same if a particular number is multiplied, added, divided, or subtracted from both sides (LHS = RHS) of the equation.
- It does not matter whether the linear equation is in one variable or two variables; it always forms a straight line when graphed.
Some characteristics of graphing in linear equations in two variables
- In the coordinate grid, every point on the scale is a solution to the linear equation.
- The solution of a linear equation in two variables will always form a point on a line in the coordinate grid.
The above explanation explains linear equations and the types of linear equations. Not just this, a simple solution to calculate linear equations in one variable and linear equations in two variables is thoroughly explained. A linear equation is used to represent data on a graph. Hence, learning how to graph it can enable understanding various mathematical concepts.