The graphical method is a visual approach used to solve a pair of linear equations in two variables. This method involves plotting the equations on a coordinate plane and identifying the point(s) where the lines intersect. The intersection point represents the solution of the equations.

1. Steps to Solve Using the Graphical Method

To solve a pair of equations graphically, follow these steps:

• Step 1: Write down the given pair of equations. For example:

$$\begin{align*} (1) \quad 2x + y &= 10 \\ (2) \quad x - y &= 1 \end{align*}$$

• Step 2: Convert each equation into slope-intercept form ($y = mx + b$) if necessary. For equation (1):

$$y = 10 - 2x$$

• For equation (2):

$$y = x - 1$$

• Step 3: Create a table of values for each equation. Choose values for $x$ and calculate corresponding $y$ values.

For equation (1):

• If $x = 0$, then $y = 10$
• If $x = 5$, then $y = 0$

For equation (2):

• If $x = 0$, then $y = -1$
• If $x = 2$, then $y = 1$
• Step 4: Plot the points on a coordinate plane:
• For equation (1): Points (0, 10) and (5, 0)
• For equation (2): Points (0, -1) and (2, 1)
• Step 5: Draw the lines for each equation using the plotted points.
• Step 6: Identify the intersection point of the two lines. This point represents the solution of the system of equations.

2. Example Problem

Consider the following pair of equations:

$$\begin{align*} (1) \quad x + 2y &= 8 \\ (2) \quad 3x - y &= 1 \end{align*}$$

Using the graphical method, we can solve these equations:

• For equation (1): Rearranging gives:

$$2y = 8 - x \implies y = 4 - \frac{1}{2}x$$

• For equation (2): Rearranging gives:

$$-y = 1 - 3x \implies y = 3x - 1$$

• Now create a table of values for both equations:

For equation (1):

• If $x = 0$, then $y = 4$
• If $x = 8$, then $y = 0$

For equation (2):

• If $x = 0$, then $y = -1$
• If $x = 1$, then $y = 2
• Plot these points on a graph and draw the lines.
• Identify the intersection point, which represents the solution.

3. Key Points to Remember

• The graphical method is useful for visualizing the relationship between equations.
• The solution corresponds to the intersection of the lines, which can indicate one solution, no solution, or infinitely many solutions depending on the nature of the equations.
• This method is best for simple equations or for understanding the concepts of linear relationships.

4. Types of Solutions

In the graphical method, three scenarios may occur:

• One Solution: The lines intersect at one point, indicating a unique solution.
• No Solution: The lines are parallel and never intersect, indicating that the equations are inconsistent.
• Infinitely Many Solutions: The lines coincide, meaning the equations represent the same line and have an infinite number of solutions.

5. Practice Problems

To strengthen your understanding of the graphical method, try solving the following pairs of equations:

• 1. $x + y = 5$ and $2x - y = 1$
• 2. $2x + 3y = 12$ and $x - y = 2$
• 3. $3x + y = 9$ and $x + 2y = 6$

By practicing these problems, you will gain confidence in using the graphical method to solve linear equations.