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Chapter 12: Problem 73

Find the slope and \(y\) -intercept of the graph of the equation. $$x-4 y=0$$

Short Answer

Expert verified
The slope of the graph of the given equation is \(\frac{1}{4}\) and the y-intercept is \(0\).

Step by step solution

01

Convert to slope-intercept form

Rearrange the initial equation \(x - 4y = 0\) to the slope-intercept form \(y = mx + b\). That's done by isolating \(y\). You end up with \(y = \frac{1}{4}x\). The coefficient of \(x\) is the slope, and the constant term is the y-intercept.
02

Identify the slope

In the equation \(y = \frac{1}{4}x\), the coefficient of \(x\) is the slope, which is \(m = \frac{1}{4}\).
03

Identify the y-intercept

There is no constant term in the equation, so the y-intercept is \(b = 0\).

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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 mathematical expressions that describe a straight line when plotted on a graph. They are fundamental in understanding coordinate geometry and are often expressed in the slope-intercept form: \( y = mx + b \).
Linear equations are important because they represent a constant rate of change between two variables.
  • The most common forms are the standard form \( Ax + By = C \) and the slope-intercept form \( y = mx + b \).
  • Each form is used for different purposes, but the slope-intercept form makes it easy to identify the slope and y-intercept.
Understanding how to convert between these forms is crucial for analyzing linear relationships.
Slope
The slope is a key feature of a linear equation and represents the steepness of the line. In the slope-intercept form \( y = mx + b \), \( m \) is the slope.
The slope can be positive, negative, zero, or undefined:
  • A positive slope means the line ascends from left to right.
  • A negative slope means the line descends from left to right.
  • A zero slope means the line is horizontal.
  • An undefined slope means the line is vertical.
In the example \( y = \frac{1}{4}x \), the slope is \( \frac{1}{4} \), meaning the line rises by 1 unit for every 4 units it runs horizontally. This indicates a gentle upward slope.
Y-intercept
The y-intercept is the point where the line crosses the y-axis. It is an important aspect because it gives the starting value of a function when the input (x-value) is zero.
In the slope-intercept form \( y = mx + b \), \( b \) represents the y-intercept.
  • If \( b \) is positive, the line crosses the y-axis above the origin.
  • If \( b \) is negative, the line crosses below the origin.
  • If \( b = 0 \), the line passes through the origin itself.
In our given equation, with the form \( y = \frac{1}{4}x \), \( b \) is zero, indicating that the line crosses the origin at the point (0, 0).
Coordinate Geometry
Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. It allows us to represent geometric shapes in numerical terms and analyze their properties using equations.
Factors like slope and intercepts are essential in coordinate geometry because:
  • They help to understand how lines are positioned on a coordinate plane.
  • They offer insight into the relationship between different geometric figures.
  • Coordinate geometry provides tools to solve problems involving distances, midpoints, and line intersections.
With the equation \( x - 4y = 0 \) converted to \( y = \frac{1}{4}x \), we see a clear example of how coordinate geometry uses algebra to explore geometric concepts. This conversion helps simplify the analysis of the line on a coordinate plane.

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Most popular questions from this chapter

Determine whether the line through \(P_{1}\) and \(P_{2}\) is parallel, perpendicular, or neither parallel nor perpendicular to the line through \(Q_{1}\) and \(Q_{2}\). $$P_{1}(7,-1), P_{2}(-4,6) ; Q_{1}(3,0), Q_{2}(-5,3)$$

Find the slope and \(y\) -intercept of the graph of the equation. $$4 x+3 y=12$$

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$x+3 y=6$$

Determine whether the line through \(P_{1}\) and \(P_{2}\) is parallel, perpendicular, or neither parallel nor perpendicular to the line through \(Q_{1}\) and \(Q_{2}\). $$P_{1}(0,1), P_{2}(2,4) ; Q_{1}(-4,-7), Q_{2}(2,5)$$

Evaluate a function. Given \(h(x)=3 x^{2}-2 x+1,\) find \(h(3)\).
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