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Chapter 6: Problem 57

Graph on a plane. $$ y=-\frac{3}{4} x+1 $$

Short Answer

Expert verified
Plot points (0, 1) and (4, -2), then draw the line through them.

Step by step solution

01

- Identify the slope and y-intercept

The given equation is in the slope-intercept form, which is written as: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = -\frac{3}{4} \) and \( b = 1 \).
02

- Plot the y-intercept

From the equation, the y-intercept \( b \) is 1. This means the graph crosses the y-axis at the point (0, 1). Plot this point on the graph.
03

- Use the slope to find another point

The slope \( m = -\frac{3}{4} \) indicates that for every 4 units you move to the right (positive x direction), you move 3 units down (negative y direction). Starting from (0, 1), move 4 units right to (4, 1) and then 3 units down to (4, -2). Plot the point (4, -2).
04

- Draw the line

Now that you have two points (0, 1) and (4, -2), draw a straight line through these points. This line represents the equation \( y = -\frac{3}{4} x + 1 \).

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Key Concepts

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

slope-intercept form
When working with linear equations, it's essential to understand the slope-intercept form. This form is written as:
$$ y = mx + b $$
Here, m represents the slope of the line, and b is the y-intercept. The slope-intercept form makes it easy to identify the slope and y-intercept, which are crucial for graphing. In the equation given in the problem,
$$ y = -\frac{3}{4} x + 1 $$
we can directly see that the slope \( m \) is \( -\frac{3}{4} \) and the y-intercept \( b \) is 1. These values are the building blocks for graphing the line.
slope
The slope of a line describes its steepness and direction. It is usually denoted by the letter m and calculated as the rise over run:
$$ m = \frac{\Delta y}{\Delta x} $$
This means for every unit you move horizontally (run), how much the line moves vertically (rise). In the example equation, the slope is \( -\frac{3}{4} \). This tells us:
  • Move 3 units down for every 4 units you move to the right.
  • The negative sign indicates a downward slope.
Understanding the slope helps in plotting additional points and understanding the direction of the line.
y-intercept
The y-intercept is where the line crosses the y-axis. This point occurs when the value of x is 0. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \). For the equation
$$ y = -\frac{3}{4} x + 1 $$
The y-intercept \( b \) is 1. This directly tells us the graph crosses the y-axis at the point (0, 1). Plotting this point is the first step in drawing our linear equation.
plotting points on a graph
To graph a linear equation, we start by plotting points. Here's a step-by-step guide based on the example:
  • Identify the y-intercept. Here, it is (0, 1).
  • Use the slope to determine another point. With a slope of \( -\frac{3}{4} \), start at (0, 1), move 4 units right to (4, 1), then move 3 units down to (4, -2).
  • Plot the second point at (4, -2).
  • Draw a line through both points.
This process ensures that the line is accurately drawn according to the equation.

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