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Chapter 4: Problem 21

Graph each equation using the slope and \(y\)-intercept. \(y+7 x=4\)

Short Answer

Expert verified
Rewrite as \( y = -7x + 4 \), plot the y-intercept (0, 4), use the slope to plot another point, and draw the line.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. To rewrite the given equation, \( y + 7x = 4 \), we need to solve for \( y \). Subtract \(7x\) from both sides of the equation to isolate \( y \): \( y = -7x + 4 \). This is now in slope-intercept form.
02

Identify the Slope and Y-Intercept

Now that the equation is in slope-intercept form \( y = -7x + 4 \), identify the slope \( m \) and the \( y \)-intercept \( b \). Here, \( m = -7 \), which is the slope, and \( b = 4 \), which is the \( y \)-intercept. This means the line crosses the \( y \)-axis at 4.
03

Plot the Y-Intercept on the Graph

Start by plotting the \( y \)-intercept on the graph. From the slope-intercept form, we know that the \( y \)-intercept is 4. Place a point on the graph at (0, 4). This is where the line crosses the \( y \)-axis.
04

Use the Slope to Plot Another Point

The slope \( m = -7 \) tells us the rise over run is -7. This can be interpreted as going down 7 units for every 1 unit you move to the right. Starting from the \( y \)-intercept (0, 4), move down 7 units and 1 unit to the right to plot another point at (1, -3).
05

Draw the Line Through the Points

With the two points plotted, (0, 4) and (1, -3), draw a straight line that passes through both points. Extend the line across the graph to complete the graph of the equation.

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Key Concepts

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

Linear Equation
A linear equation is an expression that forms a straight line when graphed. The most common form you'll encounter is the slope-intercept form, which is written as \( y = mx + b \).
This equation represents a line with two main characteristics: slope (\( m \)) and y-intercept (\( b \)). While the slope tells us how steep the line is, the y-intercept points to where the line crosses the y-axis.
Linear equations can represent real-life relationships by showing consistent change between variables.
Slope
Imagine slope as the measure of how steep a line is. When you hear slope, think rise over run. The slope \( (m) \) in our equation \( y = mx + b \) represents this concept. It's like a secret code that tells you how the line moves.
  • If \( m \) is positive, the line rises as it moves to the right.
  • If \( m \) is negative, the line falls as it moves to the right.
  • If \( m \) is zero, the line is flat.
In our exercise, the slope is -7, meaning the line falls steeply.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), \( b \) represents the y-intercept.
This point is crucial because it gives us a starting point for graphing the line.
In the given equation \( y = -7x + 4 \), the y-intercept is 4. This means that when \( x = 0 \), \( y = 4 \).
By establishing this point, you can begin plotting your line on the graph.
Graphing
Graphing involves plotting points and drawing lines on a coordinate plane. It transforms equations into visual stories.
To graph the linear equation \( y = -7x + 4 \), start by plotting the y-intercept point at (0, 4).
Next, use the slope to find another point: since the slope is -7, move down 7 units and 1 unit to the right from (0, 4) to reach (1, -3). By connecting these dots, you reveal the path of the line.
  • Always start at the y-intercept point.
  • Move according to the slope to find the next point.
  • Draw a line through these points, extending it across the graph.
Once your line is drawn, you've successfully graphed the equation, turning it into a visual representation.

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Most popular questions from this chapter

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