91Ó°ÊÓ

Slope-intercept form is a very popular way to write the equation of a line because it gives us clear information about the line's slope and where it crosses the y-axis. The general formula looks like this: \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept – the point where the line crosses the y-axis.

To convert any linear equation into this format, you need to solve for \( y \). For instance, given the equation \(-2x + 3y = 15\), you first isolate \( y \):

• Add \(2x\) to both sides to get \(3y = 2x + 15\).
• Then divide each side by \(3\) to get \(y = \frac{2}{3}x + 5\).

This equation is now in slope-intercept form with a slope of \(\frac{2}{3}\) and a y-intercept of \(5\).

This form is useful for quickly identifying how a line behaves, making it easier to predict changes and understand its relationship with other lines.

Standard form is another way of writing the equation of a line, and it has a specific structure: \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be a non-negative value.

Standard form is particularly useful in some mathematical situations such as solving simultaneous equations or identifying x- and y-intercepts quickly. For the equation \(3x + 2y = -1\), we observe:

• \( A = 3 \)
• \( B = 2 \)
• \( C = -1 \)

Here, by setting \( x = 0 \), you can directly find the y-intercept, and vice versa for the x-intercept, which can be helpful in graphing.

Converting from slope-intercept form to standard form requires rearranging the terms and ensuring you have no fractions and that the coefficient of \( x \) is positive.

When two lines are perpendicular, their slopes are negative reciprocals of each other. This means if you have the slope of one line, you can find the slope of a line perpendicular to it by flipping the fraction and changing the sign. For instance, if a line has a slope of \(\frac{2}{3}\), then the perpendicular line’s slope will be \(-\frac{3}{2}\).

This relationship helps in determining the orientation of lines relative to each other, a crucial aspect when solving geometric problems or understanding the structure of graphs. Here’s a quick way to calculate it:

• Take your initial slope, for example, \( \frac{a}{b} \).
• Flip the fraction to become \( \frac{b}{a} \).
• Change the sign, turning \( \frac{b}{a} \) to \(-\frac{b}{a} \).

This method ensures the two lines are perpendicular to each other, a unique property that maintains a 90-degree angle between them. Thus, recognizing negative reciprocals is a key mathematical skill when dealing with perpendicular lines.