91Ó°ÊÓ

Understanding how to manipulate algebraic equations is fundamental to grasp various mathematical concepts, one of which is rewriting equations into the slope-intercept form. This skill is particularly vital when dealing with linear equations. So what steps can we take to translate a general form equation, like the one from our exercise, into slope-intercept form?

To start, the slope-intercept form is given as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The goal is to isolate \( y \) on one side of the equation. Let's take the given equation \( 3x - 4y + 8 = 0 \). First, we move the \( x \)-term to the other side by subtracting \( 3x \) from both ends, which yields \( -4y = -3x + 8 \). Next, since \( y \) is multiplied by -4, we divide every term by -4, resulting in the desired form: \( y = \frac{3}{4}x - 2 \). By doing this, the equation now cleanly presents both the slope and y-intercept for easy identification.

The slope of a line is a measure of its steepness and is denoted by \( m \) in the slope-intercept form \( y = mx + b \). It can be calculated by the rise over the run between any two points on the line. In essence, it tells us how much the \( y \) value increases or decreases as \( x \) increases by one unit.

In the slope-intercept form, identifying the slope becomes a straightforward task. We simply look for the coefficient of \( x \). In our exercise, after rewriting the equation to slope-intercept form, \( y = \frac{3}{4}x - 2 \), it is clear that the slope is \( \frac{3}{4} \). This tells us that for every unit increase in \( x \), the value of \( y \) increases by \( \frac{3}{4} \) units. Understanding the slope is key to understanding how the line behaves on a graph.

The y-intercept is where a line crosses the y-axis on a graph. It's represented by \( b \) in the slope-intercept form \( y = mx + b \) and occurs when the \( x \)-value is zero. This value is particularly helpful when sketching graphs as it provides a starting point.

In the context of our exercise, the slope-intercept form obtained is \( y = \frac{3}{4}x - 2 \). Here, the y-intercept, \( b \), is the constant term and can be seen as -2. This means that the line will cross the y-axis at the point \( 0, -2 \). Recognizing the y-intercept enables us to draw an accurate representation of the line and is essential when solving graphical problems.