Problem 36 - Elementary Algebra

Question

Determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. (Objective 2) $$7 x+5 y=35$$

Step-by-Step Solution

Answer

Slope: \(-\frac{7}{5}\); y-intercept: 7.

Step 1: Identify the Function or Relation

We need to graph: \(7 x+5 y=35\).

Step 2: Determine Key Features

We identify important features such as intercepts, slope, vertex, asymptotes, or other characteristic points.

Step 3: Plot Key Points and Draw the Graph

Using the key features identified, we plot several points and connect them to form the graph.

Step 4: Describe the Graph

Slope: \(-\frac{7}{5}\); y-intercept: 7.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra. They represent straight lines when graphed on a coordinate plane. The general form of a linear equation in two variables is written as \[ ax + by = c \],where

• \(a\) and \(b\) are coefficients,
• \(c\) is a constant.

This type of equation shows the relationship between two variables, usually \(x\) and \(y\). Linear equations are ubiquitous because they model real-world situations where there is a constant rate of change or relationship between quantities. For instance, if you're driving at a constant speed, the distance you cover over time can be plotted as a linear equation. Simply put, linear equations bring order and predictability to complex systems.
To transform a linear equation into a more useful form, such as the slope-intercept form, we learn more about specific parts of the equation and how they influence the line's characteristics.

Slope

The slope of a line is a measure of its steepness or incline. In the slope-intercept formula for a line, \( y = mx + b \), the slope, \(m\), is crucial. It tells us how much \(y\) changes for a unit change in \(x\).
In our example, the given linear equation is \[ 7x + 5y = 35 \]. When rewritten in slope-intercept form, \[ y = -\frac{7}{5}x + 7 \], we see that the slope \(m\) is \(-\frac{7}{5}\).
A positive slope, like \(\frac{5}{3}\), indicates that the line rises as it moves from left to right, while a negative slope, like \(-\frac{7}{5}\), means the line falls. A larger magnitude slope indicates a steeper line. Slope provides insight into the direction and rate of the line's vertical change and is essential in predicting future points on that line.
The slope is not just a number; it is a tool that expresses a consistent rate of change across the graph.

Y-Intercept

The \(y\)-intercept is where a line crosses the \(y\)-axis. It's represented by \(b\) in the slope-intercept formula, \(y = mx + b\).In our restructured linear equation, \[ y = -\frac{7}{5}x + 7 \], the \(y\)-intercept is 7.This means that when \(x\) is 0, \(y\) equals 7.
The \(y\)-intercept provides a starting point for graphing a line. You mark the \(y\)-intercept on the coordinate plane and use the slope to determine other points on the line. The \(y\)-intercept is vital because it shows where the line will begin if you extend it from the axes.
Whether dealing with finances, physics, or any data plots, understanding where your line crosses the \(y\)-axis can offer insights into initial conditions or starting values.