91Ó°ÊÓ

The slope-intercept form is a straightforward way of writing linear equations. It is presented as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) represents the y-intercept. This form is highly convenient because it gives you both critical characteristics of the line: its steepness and where it crosses the y-axis.

For example, if you're given the slope \( m = -4 \) and the y-intercept \( b = 10 \), plugging these into the slope-intercept formula yields the equation \( y = -4x + 10 \). This method transforms two pieces of information into the complete linear equation with ease, allowing you to graph the line or make predictions about values along the line.

The slope of a line measures its steepness and direction. Mathematically, it's expressed as the ratio of the change in y (vertical change) to the change in x (horizontal change) between two points on the line. The formula is \( slope = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \). A positive slope means the line ascends from left to right, while a negative slope indicates descent.

For instance, with our example where the line passes through (2,2) and has a y-intercept at 10, we use the points (2,2) and (0,10) to find the slope. Applying the slope formula, we calculate \( slope = \frac{10 - 2}{0 - 2} = -4 \). This negative slope tells us the line goes downward as it moves from left to right across the graph.

The y-intercept is where a line crosses the y-axis on a coordinate plane. It's an essential feature because it shows the value of \( y \) when \( x \) is zero. This point is always expressed as \( (0, b) \), where \( b \) is the y-intercept value. Knowing the y-intercept is crucial for understanding where to start plotting a line on a graph and for writing equations in slope-intercept form.

In our exercise, the y-intercept is given as 10, which implies the line will cut through the y-axis at the point (0, 10). The y-intercept is a fixed point and remains unchanged regardless of the slope or any other point on the line.