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Magazine Chapter 8: Problem 10
For Problems 1-36, graph each linear equation. (Objective 2) $$ 2 x-y=-4 $$
Short Answer
Expert verified
Graph the line using points (0, 4) and (1, 6).
Step by step solution
01
Write the Equation in Slope-Intercept Form
The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start by isolating \( y \) in the equation \( 2x - y = -4 \). First, subtract \( 2x \) from both sides to get \( -y = -2x - 4 \). Then, multiply every term by \(-1\) to get \( y = 2x + 4 \). Now the equation is in the slope-intercept form.
02
Identify the Slope and Y-Intercept
In the equation \( y = 2x + 4 \), the slope \( m \) is 2, and the y-intercept \( b \) is 4. This means the line crosses the y-axis at the point (0, 4) and for every 1 unit of increase in \( x \), \( y \) increases by 2 units.
03
Plot the Y-Intercept on the Graph
Begin graphing the line by plotting the y-intercept point (0, 4) on the coordinate plane. This is where the line will cross the y-axis.
04
Use the Slope to Find Another Point
Starting at the y-intercept (0, 4), use the slope of 2, which can be written as \( \frac{2}{1} \), to find another point on the line. From (0, 4), move 1 unit to the right (increase in \( x \)) and 2 units up (increase in \( y \)), reaching the point (1, 6). Plot this second point.
05
Draw the Line
With both points (0, 4) and (1, 6) plotted, draw a straight line through these points. Extend the line across the graph to show the complete linear relationship. This line represents all possible solutions to the equation \( y = 2x + 4 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Linear Equations
Linear equations are mathematical expressions that describe a straight line when plotted on a graph. They have the general form: \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. This form can also be expressed in the slope-intercept form, \( y = mx + b \).
- "\( m \)" represents the slope of the line, which indicates how steep the line is.
- "\( b \)" is the y-intercept, which is where the line crosses the y-axis.
Graphing Linear Equations
Graphing linear equations involves plotting points and drawing a line through those points to represent the equation visually. Start by writing the equation in slope-intercept form, \( y = mx + b \), which makes it easier to identify the slope and y-intercept.
To graph:
To graph:
- Identify the y-intercept and plot it on the graph.
- Use the slope to determine another point on the line.
- Draw a straight line through the y-intercept and the second point.
Slope and Y-Intercept in Context
The slope and y-intercept are key components to understanding linear equations. The slope \( m \) tells us how the variable \( y \) changes for every unit increase in variable \( x \). In simpler terms, it describes the line's tilt.
- A positive slope means the line inclines upwards as it moves from left to right.
- A negative slope means the line declines downwards.
- The y-intercept \( b \) is the point where the line crosses the y-axis.
Exploring the Coordinate Plane
The coordinate plane is a two-dimensional space used to plot points, lines, and curves. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis, creating a grid of coordinates. Each point on the plane is represented by an ordered pair \((x, y)\).
- The x-axis represents the independent variable
- The y-axis represents the dependent variable
- Every point is measured in terms of how far it is from these two axes.
