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Chapter 1: Problem 24

Determine the slope \(m\) and \(y\) intercept \(b\) of the line with the given equation. Then sketch the line. \(x-\frac{1}{2} y=2\)

Short Answer

Expert verified
The slope is 2, and the y-intercept is -4.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of the equation of a line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To convert the given equation \(x - \frac{1}{2}y = 2\) into this form, solve for \(y\). First, isolate the \(\frac{1}{2}y\) term by subtracting \(x\) from both sides, resulting in \(-\frac{1}{2}y = -x + 2\).
02

Solve for y

Divide every term in the equation \(-\frac{1}{2}y = -x + 2\) by \(-\frac{1}{2}\) to solve for \(y\). This results in \(y = 2x - 4\). Now, the equation is in slope-intercept form, where the slope \(m = 2\) and the y-intercept \(b = -4\).
03

Identify the Slope and Y-intercept

From the equation \(y = 2x - 4\), the slope \(m\) is 2, which indicates the line rises 2 units vertically for every 1 unit it moves horizontally. The y-intercept \(b\) is -4, meaning the line crosses the y-axis at the point \((0, -4)\).
04

Sketch the Line

Begin by plotting the y-intercept \((0, -4)\) on the coordinate plane. Since the slope \(m = 2\) means \(rise = 2\) and \(run = 1\), from \((0, -4)\), move up 2 units and 1 unit to the right to find another point on the line \((1, -2)\). Draw a straight line through these points to represent the line.

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Key Concepts

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

Equation of a Line
The equation of a line is essentially a mathematical statement that tells us how points on a plane are connected. The most commonly used form is the slope-intercept form, which looks like this: \[y = mx + b\]Here:
  • \(y\) is the value of the dependent variable for any given \(x\).
  • \(m\) represents the slope, which we will discuss in detail later.
  • \(x\) is the independent variable.
  • \(b\) is the y-intercept, another core concept we'll explore.
Understanding this form is useful for quickly graphing lines and understanding their behavior. By rewriting equations into this form, you can immediately see how steep the line is and where it crosses the y-axis. This can be incredibly helpful for both solving equations and graphing them on a coordinate plane.
Slope
Slope is a crucial component of the equation of a line. It tells us how steep a line is. In the slope-intercept form \(y = mx + b\), \(m\) represents the slope. The slope is calculated as the "rise over the run," or how much \(y\) increases for each increase in \(x\).Here's how you can visualize slope:
  • If \(m = 2\), for example, it means for every unit you move to the right on the x-axis, you move up 2 units on the y-axis. This indicates a positive, upward slope.
  • A negative slope would indicate that the line is going downwards as \(x\) increases. If \(m = -1\), for instance, it means with every step to the right, you move down 1 unit.
  • A slope of 0 is a flat, horizontal line, meaning \(y\) doesn鈥檛 change as \(x\) changes.
Knowing the slope of a line helps you understand how lines interact with each other, such as whether they are parallel (having the same slope) or perpendicular (having slopes that are negative reciprocals of each other).
Y-Intercept
The y-intercept of a line is where the line crosses the y-axis on a graph. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\). This means it's simply the value of \(y\) when \(x\) is zero.Why is the y-intercept important?
  • It provides a starting point for graphing. From this point, you can use the slope to find other points on the line.
  • It shows where the line intersects the y-axis, helping to quickly sketch the line on a graph.
  • When comparing two lines, knowing their y-intercepts helps you see how they are positioned relative to each other on a graph.
In our exercise鈥檚 equation, the y-intercept \(b = -4\) indicates that the line crosses the y-axis at \((0, -4)\). This means the line has shifted downwards from the origin on the graph, relative to \(b = 0\), which would be a line passing through the point \((0,0)\).

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