Question

Write the equation in the slope-intercept form, and then find the slope and \(y\) -intercept of the corresponding lines. $$ y+4=0 $$

Short answer

The equation is already in slope-intercept form as \(y = -4\). The slope (m) is \(0\) and the y-intercept (b) is \(-4\).

Step-by-step solution

Rewrite the equation in slope-intercept form

Start by subtracting 4 from both sides of the equation to isolate y: \(y = -4\) Now, we see that the equation is already in the slope-intercept form with a slope of \(0\) (there is no x-term in the equation) and a y-intercept of \(-4\).

Identify the slope and y-intercept

From the equation in slope-intercept form (\(y = -4\)), we can see that there is no x-term, so the slope (m) is \(0\). The y-intercept (b) is the constant term, which is \(-4\). So, the slope (m) is \(0\) and the y-intercept (b) is \(-4\).

Key concepts

Equation of a Line

Understanding the equation of a line is crucial for many areas of mathematics and is particularly useful in coordinate geometry. An equation of a line provides a relationship between the x and y coordinates of any point on that particular line. One common and easy-to-use form of a line's equation is the slope-intercept form, represented as
\[ y = mx + b \].
In this notation, 'm' stands for the slope of the line—the rate at which the line inclines or declines—and 'b' stands for the y-intercept, which is the point where the line crosses the y-axis.

For instance, given the equation \[ y + 4 = 0 \], you would isolate y to arrive at the slope-intercept form, which, in this case, simplifies to \[ y = -4 \]. This represents a horizontal line that crosses the y-axis at -4. Since this form contains 'y' on one side with 'm' (the slope) and 'b' (the y-intercept) usually expressed in relation to 'x,' it is very straightforward to interpret these characteristics from the equation.

Slope of a Line

The slope of a line is a measure of its steepness and is denoted by the variable 'm' in the slope-intercept form of a line's equation. Importantly, the slope is the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line, commonly referred to as 'rise over run'.

To understand the slope in a visual context: a larger absolute value of slope indicates a steeper line, while a line with a slope of zero is perfectly horizontal. A positive slope means the line tilts upwards as it moves to the right, while a negative slope means it tilts downwards. In the solution of the exercise \[ y = -4 \], since there is no x-term, the slope is zero, implying this is a flat, horizontal line.

This concept is invaluable when you need to graph a line or predict how changing one variable affects another in a linear relationship. A real-world illustration could be comparing the driving distance (rise) with the time taken (run) to compute the average speed (slope) for a journey.

Y-Intercept

The y-intercept of a line is where it crosses or touches the y-axis of a Cartesian coordinate system. Again referring to the slope-intercept form of an equation \[ y = mx + b \], the y-intercept is represented by 'b'. This coordinate is always at the point \[ (0, b) \], since it is the output value of the function when the input value 'x' is zero.

In the context of the given exercise \[ y = -4 \], the y-intercept is \[ -4 \]. This tells us that the horizontal line meets the y-axis below the origin of the coordinate plane, at the point \[ (0, -4) \]. The concept of y-intercept is particularly useful when trying to visualize or sketch the graph of a line without plotting numerous points, as it gives an immediate place to start on the graph. It's also fundamental in many applied fields, such as economics for determining fixed costs, or in science to interpret experimental data that intercepts an axis.