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Chapter 3: Problem 2

Find the slope and the \(y\) -intercept of the line with the given equation. $$y=9 x+4$$

Short Answer

Expert verified
The slope of the line is 9 and the y-intercept of the line is 4.

Step by step solution

01

Identify the Slope

The slope is the coefficient of \(x\) in the given equation. So, by comparing the equation \(y = 9x + 4\) with the standard form \(y = mx + b\), the slope (\(m\)) is 9.
02

Identify the Y-intercept

The \(y\)-intercept is the constant term in the given equation. So, from the equation \(y = 9x + 4\), the \(y\)-intercept (\(b\)) is 4.

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Key Concepts

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

Slope of a Line
In a linear equation, the slope is crucial as it defines how steep a line is. Slope is often denoted by the letter \( m \). It's specifically expressed as the 'rise over run', meaning how much \( y \) increases or decreases as \( x \) does by one unit.
In simple terms, it's the rate of change between the variables. For the equation \( y = 9x + 4 \), the slope \( m \) is 9. This indicates two things:
  • The line goes up steeply since 9 is quite a large number.
  • For every one unit increase in \( x \), \( y \) increases by 9 units.
Recognizing the slope can help predict how the line behaves across its path on the graph.
Y-intercept
The \( y \)-intercept is another vital component in the context of linear equations. It represents the precise point where the line crosses the \( y \)-axis of the graph. In the equation \( y = 9x + 4 \), this is the constant term which is 4.
This \( y \)-intercept is easy to spot as it doesn't involve the variable \( x \). Instead, it鈥檚 the value of \( y \) when \( x \) is zero. In a practical view, when you pinpoint x at zero, you're essentially locating where the line begins on the vertical axis.
This can be visualized simply as:
  • If you imagine walking along the \( y \)-axis, you鈥檇 start at \( y = 4 \).
  • The line will cross or intercept it at this height.
Understanding the \( y \)-intercept allows us to figure out the start point of the line on a graph.
Standard Form of a Linear Equation
The standard form of a linear equation is a fundamental aspect of algebra. It is typically written as \( Ax + By = C \). While the form \( y = mx + b \), also known as the slope-intercept form, is often preferred for its clear depiction of slope and intercept, the standard form is useful in other scenarios.
Here, \( A \), \( B \), and \( C \) are integers, with \( A \) and \( B \) not both being zero. Let's explore what makes the standard form special:
  • It can be beneficial for determining or comparing different lines in a multi-line equation scenario.
  • Helps managing equations in problems involving linear combinations or intersection calculations.
Though the given equation \( y = 9x + 4 \) is in slope-intercept form, transforming it to standard form would look like \( -9x + y = 4 \). Knowing both forms enriches our mathematical versatility.

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Most popular questions from this chapter

Graph equation. \(5 y=20\)

14 is \(25 \%\) of what number?

write each sentence as a linear equation in two variables. Then graph the equation. The \(y\) -variable exceeds twice the \(x\) -variable by 5

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The ordered pair \((0,-3)\) satisfies \(y>2 x-3\)

will help you prepare for the material covered in the next section. Remember that a solution of an equation in two variables is an ordered pair. Let \(x=0\) and find a solution of \(x+2 y=0\)
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