Problem 36 - 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 -1 and passing through the point (4,3)

Step-by-Step Solution

Answer

The equation of the line is \(y = -x + 7\).

Step 1: Identify the slope-intercept form equation

The slope-intercept form of a linear equation is given by the formula \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Use the given slope

We are given the slope \(m = -1\). Substitute \(m = -1\) into the equation. Now the equation is \(y = -x + b\).

Step 3: Substitute the point into the equation

We need to find \(b\), the y-intercept. Use the point \((4, 3)\), meaning \(x = 4\) and \(y = 3\). Substitute these values into the equation \(y = -x + b\): \(3 = -(4) + b\).

Step 4: Solve for b

Simplify the equation: \(3 = -4 + b\). Add 4 to both sides to solve for \(b\): \(3 + 4 = b\). Thus, \(b = 7\).

Step 5: Write the final equation

Substitute \(b = 7\) back into the equation from Step 2. The equation of the line is \(y = -x + 7\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a fundamental representation of a linear equation. It is written as \( y = mx + b \), where:

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

This form is particularly useful because it allows us to easily graph a line and understand its behavior with just the slope and the y-intercept. For any linear equation, if you know these two values, you can describe the line completely. The formula provides a straightforward way to see what happens to \( y \) as \( x \) changes.

Slope

The slope of a line, denoted by \( m \), is a measure of its steepness. It tells us how much \( y \) increases or decreases as \( x \) increases by 1 unit. The formula to find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]The slope can be positive, negative, zero, or undefined:

• Positive slope: line rises from left to right.
• Negative slope: line falls from left to right.
• Zero slope: horizontal line.
• Undefined slope: vertical line.

In the example, we have a slope of \(-1\), which indicates the line decreases one unit in \(y\) for each unit increase in \(x\).

Y-Intercept

The y-intercept of a line is the value of \( y \) when \( x = 0 \). It is represented by \( b \) in the slope-intercept form \( y = mx + b \). In practical terms, it is the point where the line crosses the y-axis:- The y-intercept provides a starting point for graphing the line.In our example, after calculating using the point \((4, 3)\), the y-intercept \( b \) was found to be 7. Thus, the line crosses the y-axis at \( (0, 7) \). This means if you start at the origin, the line will pass through the point on the y-axis at 7.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves studying geometry using a coordinate system. It helps us understand concepts of lines and curves algebraically.

• Its foundation lies in the coordinate plane which consists of two axes: x-axis (horizontal) and y-axis (vertical).
• Each point on the plane is represented as \((x, y)\), a pair of numerical values showing its direction and distance from the origin \((0, 0)\).

In coordinate geometry, the relationship between coordinates is expressed through equations. For example, the equation of a line \( y = mx + b \) establishes a straight line by showing how \( y \) changes linearly with \( x \). In this exercise, the given point \((4, 3)\) helps determine the y-intercept when the slope is known. By substituting the point into the line's equation, we anchor the line within the coordinate plane.