Problem 35 - Brief 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 the components

The exercise has provided that the slope (m) of the line is 5 and the line passes through the point (-1, -2). To write the equation of a line, we need these two components: the slope and any point on the line.

Step 2: Substitute into point-slope form

The point-slope form equation is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the known point on the line. Substitute \( m = 5 \), \( x_1 = -1 \), and \( y_1 = -2 \) into this equation. \[ y - (-2) = 5(x - (-1)) \]

Step 3: Simplify the equation

Simplify the equation by performing the operations in parentheses and removing the negative signs. This gives:\[ y + 2 = 5(x + 1) \]

Step 4: Expand and solve for y

Distribute the 5 on the right-hand side and then solve for \( y \) to rewrite the equation in slope-intercept form \( y = mx + b \):\[ y + 2 = 5x + 5 \]Now subtract 2 from both sides:\[ y = 5x + 3 \]

Step 5: Finalize the equation

The equation of the line in slope-intercept form is now finalized. It is \( y = 5x + 3 \), where 5 is the slope and 3 is the y-intercept.

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 \( y = mx + b \), where:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, or the point where the line crosses the y-axis.

The beauty of this format is its simplicity and clarity. It immediately tells you how steep the line is (the slope) and where it crosses the y-axis.
To convert an equation into slope-intercept form, you might need to solve for \( y \) by isolating it on one side of the equation. This makes it straightforward to graph the line since you can start at the y-intercept and use the slope to determine other points on the line.

Point-Slope Form

Point-slope form offers another way to write the equation of a line, especially useful when you're given a point on the line and the slope. The general expression is \( y - y_1 = m(x - x_1) \), with:

• \( m \) as the slope.
• \((x_1, y_1)\) as a specific point on the line.

Starting from point-slope form can be easier than directly finding the slope-intercept form when you have an exact point and the slope at your disposal.
This form reveals how the line behaves relative to the specific point you know. From here, you can quickly rearrange the equation to solve for \( y \), transitioning into the slope-intercept format if desired. This makes it flexible and practical in various contexts.

Linear Equations

Linear equations represent straight lines on a graph and have a general formula of \( ax + by = c \), where:

• \( a \), \( b \), and \( c \) are constants.
• \( x \) and \( y \) are variables that depict the line on a Cartesian plane.

The term "linear" refers to the fact that the graph of these equations will always be a straight line. Linear equations are fundamental to algebra and coordinate geometry.
The simplicity of linear equations makes them a powerful tool in mathematical modeling. Whether you're dealing with economics, engineering, or physics, understanding how to manipulate and work with linear equations is essential.
Through rearrangement and substitution, these equations can be expressed in different forms, like slope-intercept or point-slope, both of which are incredibly useful depending on the information provided.