91Ó°ÊÓ

Linear equations can be expressed in the standard form, which is given by the general equation \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are constants. This form is versatile and useful for various algebraic manipulations. It's helpful when you need to discuss the properties of a line or when you want to easily rearrange the equation into different forms.
In the context of our exercise, the equation \(4x + 5y = 10\) is already in its standard form, where \(A = 4\), \(B = 5\), and \(C = 10\). Knowing the standard form aids in recognizing how to transform the equation into other useful forms like the slope-intercept form for graphing purposes. To convert it into the slope-intercept form, solve for \(y\). This conversion makes it easier to identify critical components such as the slope and y-intercept of the graph of the line.

The slope-intercept form is particularly useful for quickly identifying the key characteristics of a line: the slope and the y-intercept. This form is given by the equation \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) indicates the y-intercept.
This format is advantageous when you want to visualize the behavior of a line or understand how different coefficients affect the line's direction and position. In the exercise, after converting \(4x + 5y = 10\) to the slope-intercept form, we get \(y = -\frac{4}{5}x + 2\). Here,

• The slope \(m = -\frac{4}{5}\) shows the steepness and the direction of the line indicating that for every 5 units increase in \(x\), \(y\) decreases by 4 units.
• The intercept \(b = 2\) suggests that the point where the line crosses the y-axis is (0, 2).

This form is often used in graphing, making it simpler to understand how changes in \(x\) affect \(y\).

Graphing linear equations involves plotting points on a Cartesian coordinate system and drawing a line through them. It's a powerful visual tool allowing us to see how equations represent relationships between variables.
To graph an equation like \(y = -\frac{4}{5}x + 2\) from our exercise, follow these steps:

• Identify and plot the y-intercept, \(b\). For this line, the y-intercept is 2, so you mark the point (0, 2) on the y-axis.
• Use the slope \(-\frac{4}{5}\) to find another point. This means starting at (0, 2), you move 5 units to the right (positive x-direction) and 4 units down (negative y-direction), leading to the point (5, -2).
• Draw a straight line through these points, extending in both directions. This visually represents the linear equation and shows how y changes with x.

Understanding how to graph linear equations provides great insight into the equation's properties, such as slope and intercepts, and offers a geometric interpretation of algebraic expressions.