Question

Why is the equation \(y=m x+b\) called the slope-intercept form?

Short answer

The equation \(y = mx + b\) is called the slope-intercept form because it clearly presents the slope (\(m\)) and the y-intercept (\(b\)) of the line. The slope (\(m\)) is multiplying \(x\), and the y-intercept (\(b\)) is added to \(mx\).

Step-by-step solution

Understanding the Equation's Terms

Begin by understanding the equation \(y = mx + b\). This is a linear equation, where each term has a specific meaning: \(y\) and \(x\) are variables, \(m\) represents the slope of the line, and \(b\) is the y-intercept.

Defining the Slope

The slope \(m\) of the line is the value that indicates the steepness and direction of the line. It is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line, and is represented by the coefficient of \(x\) in the equation. In other words, a line's slope equals the change in \(y\) divided by the change in \(x\).

Defining the Intercept

The y-intercept \(b\) represents the point where the line crosses the y-axis. In other words, it's the \(y\) value when \(x = 0\). In the equation \(y = mx + b\), \(b\) describes the y-values when the value of \(x\) is zero.

Connecting the term Slope-Intercept

The equation \(y = mx + b\) is called 'slope-intercept form' because it clearly presents the slope (\(m\)) and the y-intercept (\(b\)) of the line. The slope (\(m\)) is multiplying \(x\), and the y-intercept (\(b\)) is added to \(mx\). This makes it easy to identify both the slope and the y-intercept from the equation.

Key concepts

Linear Equations

Linear equations are fundamental in mathematics and are used to represent relationships where there is a constant rate of change. They appear in the form of \(y = mx + b\), often referred to as the slope-intercept form. This equation shows a straight line when graphed on a Cartesian plane, where \(x\) and \(y\) are variables that correspond to the horizontal and vertical coordinates on the graph.

These equations are essential tools for modeling real-world situations, like calculating distance over time, predicting profits, or analyzing trends. The beauty of a linear equation lies in its simplicity; it's the mathematical shorthand for saying 'as one thing changes, another thing changes at a constant rate'. In the equation \(y = mx + b\), the \(m\) and \(b\) are not variables but constants that dictate the line’s behavior: its slope and where it intercepts the y-axis, respectively.

Slope of a Line

The slope of a line, represented by \(m\) in the equation \(y = mx + b\), tells us about the direction and steepness of the line. To visualize it, imagine climbing up a hill; the slope is how steep that hill is. A positive slope means the line goes uphill as you move from left to right, and a negative slope means it goes downhill.

Calculating the Slope

In mathematics, the slope is calculated as the ratio of the 'rise' (vertical change) over the 'run' (horizontal change) between two points on the line. The formula is given by \(m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}\), where \(\Delta\) (delta) represents a change. If you have two points on a line, \((x_1, y_1)\) and \((x_2, y_2)\), the slope is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Knowing the slope helps predict the behavior of the line and understand the relationship between variables.

Y-intercept

The y-intercept, designated as \(b\) in the slope-intercept equation \(y = mx + b\), is an essential concept in understanding how linear equations behave. It represents the point where the line crosses the y-axis, which is when \(x = 0\). At this point, the equation simplifies to \(y = b\), meaning the y-intercept is the y-value of the line when the x-value is zero.

This point can be considered the starting point of the line on the graph. It’s an integral part of the slope-intercept form because it provides an exact location on the y-axis, making it easier to graph the line. Whether you are looking at a graph or working with an equation, finding the y-intercept gives you the point from which you can 'start' the line and use the slope to continue drawing it in either direction.