Problem 10 - Algebra 1

Question

Find the slope and the y-intercept of the graph of the equation. $$ y=-x-2 $$

Step-by-Step Solution

Answer

The slope of the line is -1 and the y-intercept is -2.

Step 1: Identify the Form of the Equation

The equation \(y=-x-2\) is given. This equation is in the slope-intercept form \(y=mx+b\), where 'm' is the slope of the line and 'b' is the y-intercept.

Step 2: Find the Slope

The coefficient of 'x' in the equation represents the slope 'm'. Looking at the equation, we see that the coefficient of 'x' is -1. Therefore, the slope of the line is -1.

Step 3: Find the Y-Intercept

The constant in the equation represents the y-intercept 'b'. Looking at the equation, the constant is -2. Therefore, the y-intercept of the line is -2.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial when studying linear equations. The slope is a measure of how steep a line is, and it can be thought of as the 'rise over run'. In other words, it tells us how much the y-value changes for a unit change in the x-value. The slope is usually represented by the letter 'm'.

For example, if we have a line that rises by 2 units for every 3 units it runs to the right, the slope would be \( m = \frac{2}{3} \). The sign of the slope indicates the direction the line is moving: a positive slope means the line is inclining upwards as you move from left to right, while a negative slope indicates the line is descending. In the given equation \( y = -x - 2 \), the slope is -1, indicating a line that decreases by 1 unit vertically for each 1 unit it moves horizontally.

Y-Intercept

The y-intercept is another fundamental concept when dealing with linear equations. It signifies the point where the line crosses the y-axis. Mathematically, this is where the value of x is zero. The y-intercept is generally denoted by 'b'.

When you're given a slope-intercept form of a line \( y = mx + b \), the y-intercept can be easily read off as the constant 'b'. In our exercise, the equation is \( y = -x - 2 \), so the y-intercept is -2. This means that the line will cross the y-axis at the point (0, -2). Locating the y-intercept on a graph provides a starting point for drawing the line, especially when combined with the slope.

Linear Equations

Linear equations are the simplest type of equations you'll encounter in algebra. They describe a straight line when plotted on a coordinate system. The general form of a linear equation is \( y = mx + b \), where 'm' is the slope, and 'b' is the y-intercept. This format is known as the slope-intercept form because it directly shows the slope and y-intercept, which are key in graphing the line.

Linear equations can come in different forms, but they can often be rearranged into the slope-intercept form by solving for y. The standard form of a linear equation is \( Ax + By = C \), where 'A', 'B', and 'C' are constants. To convert this into the slope-intercept form, you would solve for y, making it easier to identify the slope and y-intercept. Essentially, understanding linear equations is about recognizing patterns and using these to graph lines or solve for variables.