Problem 20 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-1,\) passing through \(\left(-4,-\frac{1}{4}\right)\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y + 1/4 = -1(x + 4)\). After simplification, the equation of the line in slope-intercept form is \(y = -x - 4.25\).

Step 1: Use Point-Slope Form

The formula for point-slope form is \(y - y1 = m(x - x1)\), where \((x1, y1)\) is the point that the line passes through and \(m\) is the slope of the line. Substituting the given values into this equation gives us \(y - (-1/4) = -1(x - (-4))\). Simplifying this equation gives \(y + 1/4 = -1(x + 4)\).

Step 2: Simplify Equation

Through simplification, we get our final equation in point-slope form: \(y = -x - 4 - 1/4\), which can be further simplified to \(y = -x - 4.25\).

Step 3: Use Slope-Intercept Form

The formula for slope-intercept form is \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. Since we already have our equation in slope-intercept form \(y = -x - 4.25\) after simplification in step 2, we can see that the slope \(m\) is -1 and the y-intercept \(b\) is -4.25, matching what we have.

Key Concepts

Point-Slope Form

The point-slope form is one of the most commonly used forms of linear equations. It is defined by the equation \(y - y_1 = m(x - x_1)\). This form is particularly helpful when you know a specific point on a line and the slope of the line. Here's how you can use it efficiently:- **Point**: The point \((x_1, y_1)\) is any point that the line passes through. In our original exercise, this point is \((-4, -\frac{1}{4})\).- **Slope**: The slope \(m\) represents the steepness of the line. It describes how much \(y\) increases or decreases as \(x\) increases by one unit. For example, a slope of \(-1\) indicates that for every increase of 1 in \(x\), \(y\) decreases by 1.To write the equation in point-slope form, substitute the point and slope into the formula. For the original problem, we substitute to get \(y + \frac{1}{4} = -1(x + 4)\). This format is not only compact but straightforward to plug values into.Understanding point-slope form allows you to quickly establish a line’s equation and convert it to other forms if needed.

Slope-Intercept Form

The slope-intercept form is another popular form for linear equations. It is expressed as \(y = mx + b\) and is very intuitive for interpreting graphs.- **Slope \((m)\):** As mentioned before, the slope determines the line's steepness and direction. In slope-intercept form, the coefficient of \(x\) represents the slope.- **Y-intercept \((b)\):** This form specifically showcases the \(y\)-intercept, which is the point at which the line crosses the \(y\)-axis. It is represented by \(b\) in the equation.After simplifying the point-slope form equation, we arrived at \(y = -x - 4.25\). Here, \(-1\) is the slope, showing a downward line as \(x\) increases, and \(-4.25\) is the y-intercept, meaning the line crosses the \(y\)-axis at \(-4.25\). This form offers clarity in quickly identifying the behavior of the line on a graph.Using slope-intercept form allows easy graphing and understanding of linear relations at a glance.

Slope of a Line

The slope is a fundamental part of linear equations, describing the incline of a line. It is a measure of how much \(y\) changes for a unit increase in \(x\). In mathematical terms, it is often described by \(m\), calculated as \(\frac{\text{change in } y}{\text{change in } x}\).- **Positive Slope:** Implies the line rises as it moves from left to right.- **Negative Slope:** Indicates the line falls as it moves left to right. In our example, the slope is \(-1\), clearly showing a decline for every unit \(x\) increases.- **Zero Slope:** Results in a horizontal line, showing zero change in \(y\) regardless of \(x\) changes.- **Undefined Slope:** Seen in vertical lines, which do not have a defined slope because rising \(x\) values don't affect \(y\). Understanding and calculating the slope is crucial because it impacts the line's direction and relationship between \(x\) and \(y\). In the given problem, the slope \(-1\) simplifies the prediction of how the line will behave on a coordinate plane. Grasping this concept helps students easily transition between different forms of linear equations.