Problem 35 - Applied Calculus

Question

Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Slope 5 and passing through the point (-1,-2)

Step-by-Step Solution

Answer

The equation is \(y = 5x + 3\).

Step 1: Identify Key Information

Given in the problem, the line has a slope of 5 and passes through the point (-1,-2). We need to use these values to formulate the equation of the line in the form \(y = mx + b\).

Step 2: Use the Point-Slope Formula

The point-slope form of a line is given by the equation \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the point through which the line passes. Substitute \(m = 5\), \(x_1 = -1\), and \(y_1 = -2\) into the formula.

Step 3: Substitute Known Values

Plug the given information into the point-slope formula: \[ y - (-2) = 5(x - (-1)) \] This simplifies to: \[ y + 2 = 5(x + 1) \]

Step 4: Simplify the Equation

Expand the equation by distributing the 5:\[ y + 2 = 5x + 5 \] Then, isolate \(y\) by subtracting 2 from both sides:\[ y = 5x + 5 - 2 \] Thus, \[ y = 5x + 3 \].

Step 5: Final Equation

The equation of the line in the slope-intercept form is \(y = 5x + 3\). This is derived from the point-slope form and satisfies both the slope and the point condition.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to express the equation of a line. It is written as \(y = mx + b\).
In this equation, \(m\) represents the **slope** of the line, which indicates the line's steepness or incline. The variable \(b\) stands for the **y-intercept**, the point where the line crosses the y-axis.
This form is very helpful because it quickly tells you both the slope of the line and where the line starts in relation to the y-axis.

• **Easy to read**: The slope-intercept form directly reveals the slope of the line and the y-intercept.
• **Graph-friendly**: Ideal for plotting graphs since it provides a clear starting point and direction.

When writing linear equations in this form, it simplifies understanding how changes in the slope \(m\) or the y-intercept \(b\) affect the graph of the line.

Point-Slope Form

The point-slope form is another essential way to express the equation of a line. This form is notably written as \(y - y_1 = m(x - x_1)\).
Here, \(m\) is the **slope** of the line, and \((x_1, y_1)\) is a point on the line. This form is particularly useful when you know a single point through which the line passes and the slope of the line.
Using these two pieces of information, it's possible to construct the complete line equation.

• **Versatile**: Very useful when given a point and a slope.
• **Quick Conversion**: Can be easily transformed into the slope-intercept form by isolating \(y\).

By using the known point \((x_1, y_1)\) and the slope \(m\), you can insert them directly into the equation and then manipulate it if you need to convert it to the slope-intercept form later.

Slope Calculation

The slope of a line, often denoted as \(m\), is a measure of its steepness.
The most basic formula for calculating the slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points, which is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on the line. This formula essentially gives you:

• **Rise**: The difference in the y-coordinates \((y_2 - y_1)\).
• **Run**: The difference in the x-coordinates \((x_2 - x_1)\).

Essentially, the slope compares how much the line goes up or down versus how much it goes left or right.
If the slope is positive, the line ascends as you move from left to right. Conversely, a negative slope means the line descends. A zero slope indicates a perfectly horizontal line, while an undefined slope characterizes a vertical line.