Problem 17 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-4,\) passing through (-4,0)

Step-by-Step Solution

Answer

The point-slope form of the line is \(y=-4(x+4)\) and the slope-intercept form of the line is \(y=-4x-16\).

Step 1: Point-Slope Form

First, substitute the provided slope and point into the point-slope form equation. The slope \(m\) is given as -4, and the point as \(-4,0\). Substituting these into the equation \(y-y_1=m(x-x_1)\), we get \(y-0=-4(x-(-4))\) which simplifies to \(y=-4(x+4)\). This is the equation of the line in point-slope form.

Step 2: Slope-Intercept Form

To convert the equation into slope-intercept form, we want to isolate \(y\) on the left side. Simplifying the equation \(y=-4(x+4)\), we end up with \(y=-4x-16\). This is the equation of the line in slope-intercept form.

Key Concepts

Point-Slope Form

Starting with the point-slope form is a great stepping stone in writing linear equations because it uses a known point and the slope to define the line. It is written as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point on the line, and \((m)\) is the slope. When we are given a point, such as (-4, 0), and a slope, such as -4, we plug these into the equation.

Let's apply this to our exercise. We have the slope \((-4)\) and the point \((-4, 0)\). Placing these into the point-slope form gives us \(y - 0 = -4(x - (-4))\), which simplifies to \(y = -4(x + 4)\). Remember, this form is particularly useful when you need to create an equation quickly from a graph or a set of points.

Slope-Intercept Form

The slope-intercept form is one of the most commonly used representations of a line. Written as \(y = mx + b\), it clearly shows the slope \((m)\) and the y-intercept \((b)\) of the line.

To transform the point-slope equation to slope-intercept form, we simply need to isolate \((y)\) and simplify. From our previous example \(y = -4(x + 4)\), we expand and simplify to get \(y = -4x - 16\). This equation tells us that the line crosses the y-axis at -16 (the intercept) and for every one unit increase in \((x)\), \((y)\) decreases by 4 units (the slope). It's an elegant way to graph a line or to understand the line's characteristics quickly.

Linear Equations

Linear equations are the foundation for understanding algebra and represent straight lines on a graph. Their standard forms are the point-slope form and the slope-intercept form, which we've already explored. In general, a linear equation will look like \(Ax + By = C\), where \((A)\), \((B)\), and \((C)\) are constants.

The reason they are 'linear' is because the power of both \((x)\) and \((y)\) is 1, and when graphed, they result in a straight line. Linear equations are vital for modeling real-world phenomena like speed, the flow of traffic, or economic trends since many relationships can be approximated by straight lines.

Slope of a Line

The slope is a measure of the steepness of a line, and it's crucial for understanding how a line behaves. It is represented by \((m)\) in our equations and calculated by the difference in the y-values divided by the difference in the x-values between two points on the line (\((m = (y_2-y_1) / (x_2-x_1)\)).

In terms of our exercise, a slope of -4 means that for every unit increase in \((x)\), \((y)\) decreases by 4 units. Visualizing this, we would see a line that falls from left to right. Whether you're predicting future events or deciphering patterns, understanding the concept of slope is invaluable in mathematics and its applications.