Problem 18 - Algebra 1

Question

Find the slope and the y-intercept of the graph of the equation. $$ 3 x+4 y=16 $$

Step-by-Step Solution

Answer

The slope is -3/4 and the y-intercept is 4.

Step 1: Rewriting the Equation

First, let's rewrite the given equation in the form of \( y = mx + b \). In order to do that, apply basic algebraic operations to isolate y on one side of the equation. Start with \( 3x + 4y = 16 \) and subtract \( 3x \) from both sides. So, the equation becomes \( 4y = -3x + 16 \).

Step 2: Convert to Slope-Intercept Form

To manipulate the equation into the form \( y = mx + b \), divide each term of \( 4y = -3x + 16 \) equation by 4. This results in \( y = -3/4x + 4 \).

Step 3: Identify Slope and y-Intercept

In the slope-intercept form equation \( y = mx + b \), the slope \( m \) is the coefficient of \( x \), and the \( y \)-intercept \( b \) is the constant term. So, in this case, the slope (m) is -3/4, whereas the y-intercept (b) is 4.

Key Concepts

Finding Slope

The slope of a line is a measure of its steepness and direction. In algebra, finding the slope is essential for understanding the behavior of linear functions. The general formula to calculate the slope, represented as 'm', is \( m = \frac{{rise}}{{run}} \), where 'rise' represents the vertical change, and 'run' represents the horizontal change between two points on a line.

In the context of the standard slope-intercept form, \( y = mx + b \), the slope can be easily identified as the coefficient of the \(x \) variable. For our exercise \( 3x + 4y = 16 \), rewriting the equation to the slope-intercept form \( y = -\frac{3}{4}x + 4 \) reveals that the slope 'm' is \( -\frac{3}{4} \). This negative value indicates that the line is slanting downwards from left to right.

Understanding the concept of slope enables students to analyze the rate of change and compare the steepness of various lines, which is a vital skill in both mathematics and real-world applications.

y-intercept

The y-intercept is the point where a line crosses the y-axis. When dealing with linear equations, the y-intercept is incredibly useful because it gives us a starting point on the graph. In the slope-intercept form of a line, \( y = mx + b \), the y-intercept is expressed as the 'b' value.

To find the y-intercept in our exercise, we convert the original equation to the slope-intercept form, resulting in \( y = -\frac{3}{4}x + 4 \). The constant term '4' is the y-intercept, signifying that the line will cross the y-axis at the point (0,4). It's important for students to remember that the y-intercept is always the y-coordinate of the point where \(x = 0\).

In real-life scenarios, such as understanding economic models or predicting statistics, the y-intercept represents a starting value before other factors come into play and is crucial for interpreting data correctly.

Algebraic Operations

Algebraic operations are the foundation of solving equations and manipulating expressions. These operations include addition, subtraction, multiplication, division, and the use of exponents. To solve the initial exercise, we apply these basic operations to isolate the variable 'y'.

Starting with \( 3x + 4y = 16 \), we first subtract \(3x\) from both sides to get \(4y = -3x + 16\). This step uses the operation of subtraction. Then, to get the y by itself, we divide everything by '4', a division operation, resulting in the final slope-intercept form \( y = -\frac{3}{4}x + 4 \).

Mastering these algebraic operations is vital, as they are used consistently throughout mathematics to simplify expressions and solve equations. This process of isolating variables is the cornerstone of algebra and gives students the tools to uncover solutions and understand mathematical relationships.