Problem 27 - Algebra 1

Question

Find the \(y\) -intercept of the graph of the equation. $$ y=3 x-4 $$

Step-by-Step Solution

Answer

The \(y\)-intercept of the equation \(y = 3x - 4\) is \(-4\). Hence, the point at which the line crosses the y-axis is at (0, -4).

Step 1: Identify the Equation

First, recognize the equation is of the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. The given equation is \(y = 3x - 4\). From this equation, we can identify that the coefficient of \(x\) is \(3\), which is the slope, and the constant term, \(-4\), is the y-intercept.

Step 2: Instance When x=0

To calculate the \(y\)-intercept, one needs to determine the value of \(y\) when the value of \(x = 0\). Substitute \(x = 0\) into the equation. The equation will be reduced to \(y = 3*0 - 4\), which simplifies to \(y = -4\).

Step 3: Determine the y-intercept

By substituting \(x = 0\) into the equation, we have found that \(y = -4\). Hence, the y-intercept is \(-4\) and the corresponding point on the graph is \( (0, -4) \).

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra. They represent straight lines on a graph and are usually written in the form of \( y = mx + c \), where:

• \( y \) and \( x \) are variables that represent some values on a coordinate plane.
• \( m \) is the slope, indicating how steep the line is.
• \( c \) is the y-intercept, the point where the line crosses the y-axis.

Understanding linear equations is essential because they form the basis for more complex mathematical concepts. They show direct proportionality between two quantities, expressing a constant rate of change.
In the exercise example, the linear equation \( y = 3x - 4 \) is given. This shows a direct relationship between \( x \) and \( y \) with a constant slope and y-intercept, making it a straightforward representation of a line on a graph.

Coordinate System

A coordinate system is a two-dimensional plane where we can plot each point using a pair of numbers, known as coordinates. This system allows for the visualization of equations, like linear equations, in a graphical format.

• The horizontal axis is called the x-axis.
• The vertical axis is called the y-axis.
• Each point in this system is defined by an \((x, y)\) coordinate.
• The origin of the system is the point \((0, 0)\).

Using this system, you can easily visualize the behavior of linear equations. For instance, in \( y = 3x - 4 \), the y-intercept at \((0, -4)\) shows where the line will cross the y-axis. Placing points like this on a graph helps to understand how changes in \( x \) influence \( y \), embodied by the slope and y-intercept of the line.

Slope-Intercept Form

The slope-intercept form of a linear equation is a specific way of writing the equation of a line so that its characteristics are easily identifiable. It is expressed as \( y = mx + c \), where the slope \( m \) and the y-intercept \( c \) are explicitly stated.

• The slope \( m \) indicates the tilt of the line. A higher value means a steeper line.
• The y-intercept \( c \) tells you where the line will cross the y-axis.

In our example \( y = 3x - 4 \), the slope \( m = 3 \) signifies that for each unit increase in \( x \), \( y \) will increase by 3 units. The y-intercept \( -4 \) means the line crosses the y-axis at \((0, -4)\). This form makes it easy to graph the line and understand its behavior at a glance.