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Chapter 2: Problem 13

Determine the \(y\) -intercept. $$ y=5 x+3 $$

Short Answer

Expert verified
The y-intercept is 3.

Step by step solution

01

Identify the Slope-Intercept Form

Recognize that the given equation is in the slope-intercept form, which is generally written as y = mx + b.
02

Identify the y-intercept

In the slope-intercept form equation y = mx + b, the y-intercept is represented by the constant term 'b'. This is the value of y when x = 0.
03

Extract the y-intercept value

From the given equation, y = 5x + 3, identify the constant term 'b'. Here, b = 3.
04

Write the y-intercept

Therefore, the y-intercept is the point where the line crosses the y-axis, which is at y = 3.

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Key Concepts

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

slope-intercept form
The slope-intercept form is a way of writing linear equations. It is very handy for easily understanding the properties of a line.
This form of a linear equation is written as: \(y = mx + b\), where:
  • \(y\) is the dependent variable
  • \(x\) is the independent variable
  • \(m\) is the slope of the line
  • \(b\) is the y-intercept, or the point where the line crosses the y-axis
In the given exercise, the equation \(y = 5x + 3\) is already in the slope-intercept form.
Here, \(m = 5\) (the slope) and \(b = 3\) (the y-intercept).
Understanding this form helps in quickly identifying these key features of the line.
linear equations
Linear equations form straight lines when plotted on a graph.
They describe a relationship between two variables, typically \(x\) and \(y\).
To be a linear equation, it needs to be in the first degree (the highest power of the variable, like \(x\), is 1).
There are no terms with exponents higher than 1, no products of variables, or other funky stuff.
Examples of linear equations include:
  • \(y = 2x + 1\)
  • \(y = -3x + 4\)
  • \(y = 0.5x - 7\)
The equation in the exercise, \(y = 5x + 3\), shows that as well.
It's straightforward and linear because the highest power of \(x\) is just 1, so the graph will be a straight line.
constant term
The constant term in a linear equation in slope-intercept form is the \(b\) in \(y = mx + b\).
It is called the constant term because it does not change. It has no variable attached to it.
This term represents the y-intercept, which is the point where the line crosses the y-axis.
To find the y-intercept, set \(x = 0\) and solve for \(y\).
In our exercise example, \(y = 5x + 3\), the constant term is 3.
This means if you were to plug \(x = 0\) into the equation, \(y = 5(0) + 3\), it shows \(y = 3\).
Therefore, the line crosses the y-axis at \(y = 3\). This point is also written as (0, 3).

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Most popular questions from this chapter

Write an equation, in standard form, for the line whose \(x\) -intercept is 5 and whose \(y\) -intercept is \(-4\)

Using the window \([-5,5,-1,9],\) graph \(y_{1}=5\) \(y_{2}=x+2,\) and \(y_{3}=\sqrt{x} .\) Then predict what shape the graphs of \(y_{1}+y_{2}, y_{1}+y_{3},\) and \(y_{2}+y_{3}\) will take. Use a graphing calculator to check each prediction.

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ \frac{1}{2}(x-4)=y $$

Solve graphically and then check by solving algebraically. $$ 4 x-1=3-2 x $$

Evaluate \(9 x y \div z^{2} x\) for \(x=-2, y=6,\) and \(z=3\)
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