Problem 20 - 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 \(-1\) and \(y\) -intercept \((0,5)\)

Step-by-Step Solution

Answer

The equation of the line in standard form is \( x + y = 5 \).

Step 1: Use the Point-Slope Form

We start by using the point-slope form of the equation of a line, which is \( y - y_1 = m(x - x_1) \). Here, \(m\) is the slope, and \((x_1, y_1)\) is a point on the line. Given the slope \(m = -1\) and the \(y\)-intercept as the point \((0, 5)\), we substitute: \( y - 5 = -1(x - 0) \).

Step 2: Simplify the Equation

Next, we simplify the equation from Step 1. We get \( y - 5 = -x \). Adding 5 to both sides gives \( y = -x + 5 \).

Step 3: Convert to Standard Form

The standard form of a line is \( Ax + By = C \). From \( y = -x + 5 \), we add \(x\) to both sides to get \( x + y = 5 \).

Step 4: Verify in Standard Form

Check the equation to ensure it meets the standard form requirement where \(A, B,\) and \(C\) are integers. Our equation \( x + y = 5 \) fits this criterion with \(A = 1\), \(B = 1\), and \(C = 5\).

Key Concepts

Point-Slope Form

The point-slope form is one way to describe how a line appears on a graph. It's great for finding an equation of a line when you know the slope and a point on the line. The general formula for the point-slope form is:

• \( y - y_1 = m(x - x_1) \)

Here, \(m\) represents the slope of the line, and \((x_1, y_1)\) is any point on the line. You can think of it as capturing how quickly the line rises or falls as you move from left to right.
In our exercise, we know the slope is \(-1\) and our known point is the y-intercept \((0, 5)\). This means we're looking at how the line changes from this specific starting point. By substituting into the formula, we get:

• \( y - 5 = -1(x - 0) \)

This version gives us a direct view of where the line crosses through \((0, 5)\) and its downward slope.

Slope-Intercept Form

The slope-intercept form is another common way to represent a line, particularly useful when you know or want to easily identify the slope and the y-intercept. The formula is:

• \( y = mx + b \)

Where \(m\) is the slope and \(b\) is the y-intercept, the point where the line crosses the y-axis.
In this exercise, after simplifying from the point-slope form, we found:

• \( y = -x + 5 \)

Here, it's clear: the slope \(-1\) makes the line decline from left to right, and the y-intercept \(5\) tells us the line crosses the y-axis where \(y = 5\). This form is incredibly visual and can be quickly sketched on a graph.

Standard Form of a Line

The standard form of a line gives an alternative representation of a linear equation, which is particularly neat and clean. This form is:

• \( Ax + By = C \)

In this formula, \(A\), \(B\), and \(C\) are integers, and the equation is often preferred for mathematical operations, such as finding intersections or using elimination in systems of equations.
From the slope-intercept form: \( y = -x + 5 \), converting to the standard form gives:

• \( x + y = 5 \)

Here, we've rearranged so all terms involving the variables are on one side and the constant on the other, with integers as coefficients. This format provides a different viewpoint but essentially describes the same line as the other forms. It's a versatile way to approach problems involving line equations.