91Ó°ÊÓ

Linear equations are mathematical expressions that represent a straight line when graphed. They typically have two variables, often x and y, and can be written in the form of \(ax + by = c\). To solve a linear equation means to find the values of the variables that satisfy the equation.

Let's look at the equation from the exercise: \( x + 2y = 5\)
To solve for one variable, substitute the other variable with a specific value.

• When \(x = 0\), substitute it into the equation to find y: \(0 + 2y = 5\)
• Solve for y: \[2y = 5\]so \[y = \frac{5}{2}\]

After finding the values for one variable, you can switch and solve for the other variable by substituting a specific value into the equation and solving for the remaining variable. This method is essential for completing tables and graphing linear equations. Understanding how to solve linear equations will help you plot these points correctly.

Plotting points on a graph helps visually represent the solutions of a linear equation.
Each point represents an (x, y) pair that satisfies the equation.

• Find values for x and y by solving the linear equation for specific values as shown in the exercise.
• Use these calculated values (0, \( \frac{5}{2} \)), (5, 0), (2, \( \frac{3}{2} \)), and (1, 2) to plot points on a coordinate plane.

To plot a point, find its x-coordinate on the horizontal axis, then find its y-coordinate on the vertical axis and mark the spot where these two values intersect. Once all points are plotted, you can draw a straight line through them to represent the equation. The accuracy of these points ensures that the graphical representation of the equation is correct.

Completing a table involves filling in the missing values of x and y that satisfy the given equation.
Given a linear equation, you can systematically solve for missing values to fill out the table:

• Start by substituting a known value for x or y into the equation and solve for the other variable.
• For example: Start with \(x = 0\), substitute it into the equation \( x + 2y = 5\):
  \(2y = 5\), resulting in \(y = \frac{5}{2} \)

This process is repeated for each row of the table, substituting and solving as follows:

• When \(y = 0\): \(x = 5\)
• When \(x = 2\): \(y = \frac{3}{2}\)
• When \(y = 2\): \(x = 1\)

By systematically solving for these values, you ensure that the table is correctly completed, providing the necessary points for accurate plotting and graphing of the linear equation.