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Chapter 4: Problem 27

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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 \(y = -4\).
03

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 \( (0, -4) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

SCHOOL BAKE SALE You have one hour to make cookies for your =school bake sale. You spend 20 minutes mixing the dough. It then takes 12 minutes to bake each tray of cookies. If you bake one tray at a time, which model can you use to find how many trays you can bake? (Review 3.3 and 3.6) A. \(x(20+12)=60\) B. \(12 x+20=60\)

Find the slope of the graph of the linear function \(f\). $$ f(9)=-1, f(-1)=2 $$

You get paid $152.25 for working 21 hours. Find your hourly rate of pay.

Find the \(x\) -intercept and the \(y\) -intercept of the graph of the equation. $$3 y=6 x+3$$

Evaluate the expression for the given value of the variable. (Review 1.3 and 2.5 for 4.2 ) $$-3 x-9 y \text { when } x=-2 \text { and } y=-1$$
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