Problem 14 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line described below. $$ m=1, b=-1 $$

Step-by-Step Solution

Answer

The slope-intercept form of the given line is \(y = x - 1\)

Step 1: Understand the slope-intercept form

The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept (the value of \(y\) when \(x = 0\)). This form is useful because it allows you to quickly identify these two key properties of a line from its equation and draw it easily if necessary.

Step 2: Substitute the given values

We're given that the slope \(m = 1\) and the y-intercept \(b = -1\). So the equation becomes \(y = 1x - 1\)

Step 3: Simplify the equation

Simplify the equation by writing \(1x\) as simply \(x\). So the final equation is \(y = x - 1\)

Key Concepts

Linear Equations

Linear equations are the bedrock of algebra and represent the relationship between two variables, typically x and y, in the form of a straight line on a graph. The most commonly used form of a linear equation is the slope-intercept form, which is written as \(y = mx + b\). In this notation, \(m\) stands for the slope and \(b\) represents the y-intercept. A crucial aspect of a linear equation is its ability to model a consistent, proportional relationship between x and y.

When faced with an equation like \(m=1, b=-1\), identifying \(m\) and \(b\) allows you to shape the equation appropriately. In our case, the slope \(m\) is 1, which indicates that for each unit increase in \(x\), \(y\) increases by the same amount. The y-intercept \(b\) being -1 tells us that the line crosses the y-axis one unit below the origin.

Remember, slopes can be positive or negative, where a positive slope ascends from left to right, while a negative slope descends in the same direction, indicating the 'steepness' and direction of the line.

Graphing Lines

When it comes to graphing lines, the process involves plotting the line represented by the equation on a coordinate plane. The slope-intercept form makes this process straightforward. Firstly, start by plotting the y-intercept point on the graph, which is the point where the line crosses the y-axis. In our example, this point would be \(0, -1\).

Following this, use the slope \(m\) to determine the direction and steepness of the line. With a slope of 1, for every unit you move upwards or downwards on the y-axis, you also move one unit right or left on the x-axis, respectively. By plotting another point using the slope and drawing a line through the two points, you have graphed the linear equation.

Visualizing the Slope

If the slope is fractional, like \(\frac{2}{3}\), move up 2 units on the y-axis for every 3 units you move right on the x-axis. It's the rise over run concept, which is central to understanding how the graph of the line behaves.

Slope and Y-Intercept

The slope and y-intercept are crucial components for understanding the dynamics of a line. The slope, represented by \(m\), defines the rate of change of y with respect to x. In other words, it shows how fast or slow y is changing as x increases. A larger absolute value of the slope means a steeper line, and vice versa.

The y-intercept, denoted by \(b\), is the point where the line meets the y-axis. It's an essential aspect because it provides a starting point for plotting the line and represents the value of \(y\) when \(x=0\).

Interpreting Slope and Y-Intercept

In practical terms, if you're dealing with a line that describes a real-world scenario, the slope could represent a rate, like speed, while the y-intercept could represent a starting point, like the initial distance from a point. By understanding both \(m\) and \(b\), you can predict and interpret behaviors and outcomes in various situations, not just in mathematics but also in daily life scenarios where direct relationships between two factors are present.