Graphing Linear Equations

Graphing Linear Equations

Graphing Linear Equations is the process of representing linear equations with one or two variables on a graph. A linear equation is an equation of degree one i.e. the highest power or exponent value of the variable can only be 1, not greater than 1 in any of the cases. Solving a linear equation is to find the value of variables contained in it and the graphical method is one of the methods to solve linear equations, either one or two variables linear equations.

What is Graphing Linear Equations?

Linear equations are algebraic equations in which each term has a real constant and the equation contains 2 variables of the highest power 1. We represent the linear equation in y=mx+b form, also known as the y-intercept form. The representation of a linear equation on a graph is called graphing linear equations shown as a straight line with one or two variables. Let us see an example of graphing a linear equation with one variable. We have to represent the equation x+2y=7 in a graph.

Here, the equation x+2y=7 makes a straight line on the graph. Similarly, all linear equations create a straight line on the graph with both one or two variables. The graph of a linear equation in one variable x forms a vertical line that is parallel to the y-axis and vice-versa, whereas, the graph of a linear equation in two variables x and y forms a straight line. The graphing of linear equations helps in solving many real-life problems in linear programming.

Note:

• The point where any line crosses the x-axis on the graph – X-intercept
• The point where any line crosses the y-axis on the graph – Y-Intercept

Graphing Linear Equations

Graphing a linear equation is about solving the linear equations and representing the solution in a coordinate plane. While plotting the equation on a graph, the two pairs (x,y) are sufficient. However, we cannot find out if there are any mistakes in obtaining these values as the two points can always be joined and represented as a line. Hence, it is advised to plot one more point to ensure that the solutions obtained for the given linear equation are correct. The following steps are to be taken up for plotting a linear equation with one variable:

• Make sure the linear equation is in y-intercept form, which is y = mx + b.
• Apply the trial and error method and find the value of (x, y) up to three pairs, which satisfy the linear equation.
• Find the x-intercept and y-intercept of the equation. For y-intercept, substitute the value of x = 0 in the equation. This results in x = a, for x-intercept, substitute the value of y = 0 in the equation. This results in y = c.
• Thus, the points are (a, 0) and (0, c). Make a tabular form and put the value of x and y respectively.
• Plot all the points on the graph paper.
• Join all the points which are marked on the graph and you will get a straight line representing the given linear equation graphically.

Example: Draw a graph of the linear equation x+2y=7.

Solution: We will follow the following steps:

• Step 1: Check if the given linear equation x+2y=7 is of the form of y = mx + b. [On converting, we get: y = – (1/2) x + 7/2]
• Step 2: Find the x and y-intercept respectively. For that, put y = 0 in the equation: x = 7-2(0), x=7. Now, put x=0 in the equation. 2y=7-(0), y=7/2 = 3.5
• Step 3: Apply the trial and error method and find 3 pairs of values of (x, y) that satisfy the given linear equation x=7-2y. (See table below)
• Step 4: Plot the points (7,0), (5,1), and (3,2) on the graph.
• Step 5: Join all the points which are marked on the graph paper and get a straight line that represents the given linear equation graphically.

See the values of x and y in the following table:

Graphing Linear Equations in Two Variables

Graphing linear equations in two variables is done in a similar manner as one variable. The lines plotted on the graph can either intersect each other at one point or be parallel to each other resulting in no solution. Sometimes, the lines might coincide with each other leaving each point on that line as a solution resulting in the given system having an infinite number of solutions. If the system has a solution, then it is said to be consistent; otherwise, it is said to be inconsistent. The following are the steps to graphing linear equations in two variables:

• Step 1: To graph a linear equation with two variables, we graph both equations.
• Step 2: To graph an equation manually, first convert it to the form y=mx+b by solving the equation for y.
• Step 3: Start putting the values of x as 0, 1, 2, and so on and find the corresponding values of y, or vice-versa.
• Step 4: Identify the point where both lines meet.
• Step 5: The point of intersection is the solution.

Example: Find the solution of the following system of equations graphically.

-x+2y-3 =0

3x+4y-11=0

Solution: Using the steps mentioned above, we will graph them and see whether they intersect at a point. As you can see below, both lines meet at (1, 2). Thus, the solution of the given system of linear equations is x=1 and y=2.

Important Points

• Linear equation in two variables has infinitely many solutions.
• The graph of a linear equation is always a straight line.
• The equation y = mx is always passing through the origin (0, 0).

Challenging Questions

• The sum of the digits of a two-digit number is 8. When the digits are reversed, the number is increased by 18. Find the number.
• Jake’s piggy bank has 11 coins (only quarters or dimes) that have a value of $1.85. How many dimes and quarters does the piggy bank have?

Topics Related

Graphing Linear Equations Examples

1. Example 1: Isabella has the linear equation x – 2y = 2. Help her to draw the linear equation on the graph.

   Solution:

   The given linear equation is x-2y=2. Convert the equation in the form of y = mx + b⇒ y= x/2 – 1. We need to find the x and y-intercept respectively. For that, put y=0 in the equation⇒ x=2(0)+2, x=2. Now, put x=0 in the equation. 2y=(0) – 2, y=1. Now, we will apply the trial and error method, and find 3 pairs of values of (x, y) that satisfy the given linear equation y=x/2-1. See the values of x and y in the following table:

   x 2 4 0 y 0 1 -1

   Plot the points (2,0),(4,1),(0,-1) on the graph. Join all the points which are marked on the graph paper and get a straight line that represents the given linear equation graphically.

2. Example 2: William wants to plot the graph of -1/2x+1/3y=1. Help him to plot the graph for linear equations.

   Solution:

   The given linear equation is -1/2x+1/3y=1. Convert the equation in the form of y = mx + b⇒ 6 + 6x/2. We need to find the x and y-intercept respectively. For that, we can apply the hit and trial method, put y=6 in the equation⇒ x=3. Now, put x=2 in the equation. y=3. See the values of x and y in the following table:

   x 2 0 y 6 3 Solutions (2,6) (0,3)

Practice Questions on Graphing Linear Equations

FAQs on Graphing Linear Equations

How Do you Graph a Linear Equation?

The basic methods of the graphing linear equation:

• The first method is by plotting all points on the graph and then drawing a line through the points.
• The second is by using the y-intercept of the equation and slope of the equation.

What Are the 3 Ways to Graph a Linear Equation?

The 3 ways to graph a linear equation:

• Using two points to plot the graph of a linear equation.
• Use the slope and y-intercept of a linear equation.
• Using the x- and y-intercepts of a linear equation.

How Do you Graph a Linear Equation with Two Points?

Graphing linear equation with two points:

• Find the y-intercept from the linear equation and plot the point.
• From the y-intercept of the linear equation, use the slope to find the second point and plot it on the graph.
• Draw a straight line to connect the two points on the graph.

What Is the Formula For a Linear Equation?

The formula for a linear equation is y = mx + b.

How Do you Graph a Linear Equation Using Intercepts?

To find intercepts algebraically, we use the fact that all x-intercepts have y=0 and all y-intercepts have x=0.

Determine the corresponding values of x and y by putting the values of x-intercepts and y-intercepts respectively.

What Is the Minimum Number of Points Needed to Graph a Linear Equation?

Two points are the minimum number of points needed to graph a linear equation

How Do you Find the Y-Intercept of a Graph?

The y-intercept of a graph is the point at which the graph crosses the y-axis and at y point, the x-coordinate is zero.

How Do you Find the Intercepts of a Graph?

First of all, an equation must satisfy ax+by=c. Then you just set x = 0 to find the y-intercept and set y = 0 to find the x-intercept. Then find the corresponding values.

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