Graphing Linear Equations

Graphing linear equations means plotting them on a coordinate plane, where every point on the line represents a solution to the equation. Linear equations have the highest degree of 1 and are written in the form y = mx + b, also known as the y-intercept form.

• The graph of a linear equation with one or two variables is always a straight line, and every point on the line represents a solution of the equation. 
   
• The point where the line crosses the x-axis is called the x-intercept, and the point where it crosses the y-axis is called the y-intercept. 

Graphing a linear equation involves the process of finding its solutions and displaying them on the coordinate plane. Usually, two points (x, y) are used to plot the graph. Follow these steps to plot linear equations:

1. Rewrite the equation in slope-intercept form, that is, y = mx + b.
    
2. Choose simple x-values to find corresponding y-values to form coordinate pairs (x, y). These solutions will be represented as coordinate pairs (x, y).
    
3. Find the value of x and y intercepts by substituting y = 0 and x = 0, respectively. The points will be (0, y) and (x, 0). 
    
4. Arrange the selected x values and their corresponding y values in a table.
    
5. Plot the points on the graph and join all the points with a straight line to represent the linear equation. 
    

Practice Problem: Plot y - 3x = 2 on graph
Explanation: 

1. Slope intercept form: y = 3x + 2
    
2. Finding values of y for x = 0, 1, -1, 2
    

   x	y
   0	y = 3(0) + 2 = 2
   1	y = 3(1) + 2 = 5
   -1	y = 3(-1) + 2 = 1
   2	y = 3(2) + 2 = 8

3. Plot these points on the graph and join them to plot the graph.
    

A linear equation in two variables has the form ax + by = c or in the slope-intercept form (y = mx + b), where x and y are variables and a, b, and c are real numbers. To graph a linear equation in two variables, follow these steps: 

1. First, rewrite the equation in the form y = mx + b to identify the slope and the y-intercept
    
2. Find two or three solutions by giving different values to x and find the corresponding values of y. 
    
3. Then find the x and y intercepts
    
4. Plot the points on the graph, and connect them using a straight line. 

Practice Problem: Plot 3x + 2y - 6 = 0 on graph.

Explanation:

1. Convert the given equation into slope intercept form:
   y = (-3x + 6)/2
    
2. Find the values of y.
    

   x	y
   0	y = (0x + 6)/2 = 3
   2	y = (-3(2) + 6)/2 = 0
   1	y = (-3(1) + 6)/2 = 3/2

3. Plot the points on the graph and join them.

Students may find graphing linear equations difficult in the beginning. To make this easy, here are a few tips and tricks:

1. Always express the equation in slope intercept form. Children can use the y-intercept calculator to find the slope intercept form of a linear equation.
    
2. Arrange the obtained values of x and y in a table to avoiding making incorrect pairs of coordinates.
    
3. Be careful when computing values of y, for avoiding calculation errors.
    
4. Remember for plotting a point (a, b), a is taken on the x-axis and b on the y-axis.
    
5. Always make sure the lines passing through the points are straight.
    

Parent Tip:

• You can use graphing linear equations calculator to check your child's plotted graph.

• Encourage your child to practice problems of graphing linear equations from the given worksheet.

Learning to graph linear equations helps students solve problems practically. It is used to track expenses, predict profits, and analyze scientific data. Let’s learn some applications of graphing linear equations.

1. Graphing linear equations helps in budgeting and financial planning by visually tracking income, expenses, and savings over a period. 
   
    
2. In physics, linear equations are used to describe equations like constant motion.
   For example, if the car goes at a speed of 50 km/h, the distance traveled can be modeled by y = 50x, where x is the time in hours and y is the distance in kilometers. 
   
    
3. In business, we use linear equations to model costs, such as fixed costs and variable costs, to predict profits. Graphing helps to visualize the numbers effectively.
   
    
4. Engineers use linear equations to model structures like beams, ramps, or roads. For example, to design the wheelchair ramps to ensure safe path is done using linear relationship.
   
    
5. Temperature conversions between different units is also an example of linear equations. Graphing these linear equation gives a better visualization of temperature for data collection or science based study.