Graph of a Linear Equation in Two Variables

The graph of a linear equation in two variables is the set of all points (x,y) whose coordinates satisfy the equation.

Equation:

\[ 2x - y = 4 \implies y = 2x - 4 \]

Step-by-Step Solution:

Step 1: Write the equation and make \[ y \] the subject

\[ y = 2x - 4 \]

Step 2: Choose values of \[ x \] and find corresponding values of \[ y \]

If \[ x = 0, \] then \[ y = 2(0) - 4 = -4 \rightarrow \] Point: \[ (0, -4) \]

If \[ x = 2, \] then \[ y = 2(2) - 4 = 0 \rightarrow \] Point: \[ (2, 0) \]

If \[ x = 3, \] then \[ y = 2(3) - 4 = 2 \rightarrow \] Point: \[ (3, 2) \]

If \[ x = -1, \] then \[ y = 2(-1) - 4 = -6 \rightarrow \] Point: \[ (-1, -6) \]

Step 3: Make the Table of Ordered Pairs

\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 2 & 3 & -1 \\ \hline y & -4 & 0 & 2 & -6 \\ \hline (x, y) & (0, -4) & (2, 0) & (3, 2) & (-1, -6) \\ \hline \end{array} \]

Step 4:

Choose a scale: On both axes, \[ 1 \text{ cm} = 1 \text{ unit}. \]

Plot the ordered pairs \[ (0, -4), (2, 0), (3, 2), \] and \[ (-1, -6) \] on the coordinate plane.

Draw a straight line passing through all the points, extending it on both sides.

• Although mathematically two points are sufficient to determine a line, four points are utilized to ensure calculation accuracy.

• Plotting four points serves as a built-in verification method to detect any potential plotting or calculation errors.

• If all points lie on a single straight line, it confirms that the calculations are correct.

• A linear equation in two variables has the form \[ ax + by + c = 0, \] where \[ a \] and \[ b \] are not both zero.
• Its graph is always a straight line.
• Every solution is an ordered pair \[ (x, y). \]
• Prepare a table of values before plotting.
• Plot at least three points to verify accuracy.
• \[ y = k \] is a horizontal line parallel to the X-axis.
• \[ x = k \] is a vertical line parallel to the Y-axis.
• A line through the origin has the form \[ ax + by = 0. \]