Problem 23 - Calculus for Biology and Medicine

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope \(-2\) and \(x\) -intercept \((1,0)\)

Step-by-Step Solution

Answer

The equation of the line in standard form is \(2x + y = 2\).

Step 1: Understand the Problem

The problem asks for the equation of a line with a given slope and an x-intercept. Specifically, the slope is \(-2\) and the x-intercept is \((1,0)\). The equation should be in standard form \(Ax + By = C\).

Step 2: Use the Point-Slope Form

Using the point-slope formula, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (1, 0)\) and \(m = -2\), substitute these values to get: \[ y - 0 = -2(x - 1) \] which simplifies to: \[ y = -2x + 2 \]

Step 3: Convert to Slope-Intercept Form

The equation from the previous step \(y = -2x + 2\) is in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 4: Rearrange to Standard Form

Convert the slope-intercept form \(y = -2x + 2\) to standard form \(Ax + By = C\) by rearranging terms: Add \(2x\) to both sides: \[ 2x + y = 2 \]This is the standard form of the equation.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a popular way to express a line and is written as \( y = mx + b \). Here, the letter \( m \) stands for the slope of the line. The slope indicates how steep the line is. A positive slope means the line ascends as it moves from left to right, while a negative slope means it descends. The letter \( b \) is the y-intercept, the point where the line crosses the y-axis. This form is particularly useful when you need to quickly identify both the slope and the y-intercept.
The slope-intercept form makes it easy to graph linear equations because you can start at the y-intercept \( (0, b) \) on the graph and use the slope \( m \) to determine another point. For instance, a slope of \( -2 \) means that for every one unit you move to the right along the x-axis, you go down two units on the y-axis.

Standard Form

The standard form of a linear equation is expressed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should ideally be a non-negative number. This form is advantageous when you need to analyze the line in relation to both the x and y axes or when dealing with systems of equations.
To convert from slope-intercept form to standard form, as shown in the original problem, you'll rearrange the equation to get all variable terms on one side:

• Start with the slope-intercept form: \( y = -2x + 2 \).
• Add \( 2x \) to both sides to eliminate the x-term from the right side: \( 2x + y = 2 \).

This results in an equation in standard form, which can be more suitable for solving certain types of problems, such as finding x- or y-intercepts.

Point-Slope Form

Point-slope form is written as \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope, exactly like in the original problem. When you know a specific point on a line and the line's slope, this form is very handy.
The point-slope form is particularly useful when you're given or already know a point and the slope, because you can quickly construct the line's equation. In the sample problem:

• Take the point \( (1,0) \) and slope \( -2 \).
• Substitute into the formula: \( y - 0 = -2(x - 1) \).

This simplifies step-by-step into the slope-intercept form, \( y = -2x + 2 \), before converting to standard form. Using point-slope form can make it much easier to transition into either the slope-intercept form or standard form.