Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 8: Problem 31
For Problems 1-36, graph each linear equation. (Objective 2) $$ 3 x-4 y=7 $$
Short Answer
Expert verified
Rearrange to slope-intercept form: \( y = \frac{3}{4}x - \frac{7}{4} \), then graph.
Step by step solution
01
Rearrange the Equation
To graph the linear equation, first rearrange it into the slope-intercept form, which is \( y = mx + b \). Starting with the equation \( 3x - 4y = 7 \), solve for \( y \): Subtract \( 3x \) from both sides:\[ -4y = -3x + 7 \]Now, divide everything by \(-4\) to solve for \( y \):\[ y = \frac{3}{4}x - \frac{7}{4} \].
02
Identify the Slope and Y-Intercept
From the equation \( y = \frac{3}{4}x - \frac{7}{4} \), identify the slope \( m \) and the y-intercept \( b \). Here, the slope \( m = \frac{3}{4} \) and the y-intercept \( b = -\frac{7}{4} \).
03
Plot the Y-Intercept
On the coordinate plane, plot the y-intercept. Since the y-intercept \( b = -\frac{7}{4} \), plot the point \( (0, -\frac{7}{4}) \).
04
Use the Slope to Find Another Point
The slope \( \frac{3}{4} \) means for every increase of 3 in \( x \), \( y \) increases by 4. Starting at \( (0, -\frac{7}{4}) \), move right 4 units and up 3 units to find a second point \( (4, -1) \).
05
Draw the Line
With the points \( (0, -\frac{7}{4}) \) and \( (4, -1) \) plotted, draw a straight line through these points to graph the equation \( 3x - 4y = 7 \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Linear Equations
Graphing linear equations involves plotting points on a coordinate plane and connecting them with a straight line. A linear equation is one that can form a straight line when graphed. These equations typically take the form of \( y = mx + b \) or are easily rearranged into this format. By identifying the slope and y-intercept, we can accurately sketch the line that represents the equation.
When graphing:
When graphing:
- Start by rewriting the equation in a form suitable for plotting, typically the slope-intercept form.
- Next, plot the y-intercept on the y-axis.
- Use the slope to determine another point on the line.
- Finally, draw a line through these points, extending in both directions.
Slope-Intercept Form
The slope-intercept form is a way of writing linear equations and is expressed as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept. This form is popular because it clearly shows these two essential characteristics of the line.
To rewrite an equation in this form:
To rewrite an equation in this form:
- Isolate \( y \) on one side of the equation using algebraic operations.
- The coefficients and constants will give you the slope and y-intercept directly.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. In the equation \( y = mx + b \), the y-intercept is \( b \). This is a crucial point because it helps anchor the line on the graph.
To find the y-intercept:
To find the y-intercept:
- Set \( x = 0 \) in the equation and solve for \( y \). This will give you the y-coordinate of the intercept.
- On a graph, the y-intercept is represented by the point \( (0, b) \).
Coordinate Plane
A coordinate plane is composed of two intersecting perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants and are used to plot points identified by pairs of numbers (coordinates), \( (x, y) \).
When working with a coordinate plane:
When working with a coordinate plane:
- The point \( (0, 0) \) is known as the origin, where the axes intersect.
- The x-coordinate represents horizontal positioning, and the y-coordinate represents vertical positioning.
- Points are plotted using these coordinates, and a line is drawn through connected points to graph a linear equation.
