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Magazine Chapter 8: Problem 29
For Problems 1-36, graph each linear equation. (Objective 2) $$ -2 x+y=-4 $$
Short Answer
Expert verified
Graph the line with y-intercept -4 and slope 2.
Step by step solution
01
Identify the Slope-Intercept Form
We need to graph the equation, and a good starting point is to write it in slope-intercept form, which is \( y = mx + b \). The given equation is \( -2x + y = -4 \).
02
Solve for y
Rearrange the equation to solve for \( y \). Add \( 2x \) to both sides to isolate \( y \): \( y = 2x - 4 \). Now the equation is in slope-intercept form.
03
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 will rise 2 units vertically for every 1 unit it moves horizontally, and it crosses the y-axis at -4.
04
Plot the Y-Intercept
Plot the point (0, -4) on the graph. This is where the line crosses the y-axis.
05
Use the Slope to Find Another Point
Starting from the y-intercept (0, -4), use the slope to find another point. From (0, -4), move up 2 units and right 1 unit to reach the point (1, -2). Plot this point.
06
Draw the Line
Draw a line through the two points (0, -4) and (1, -2). Extend this line across the graph to represent the linear equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
When graphing a linear equation, the first step is often to get the equation into slope-intercept form. This format is \( y = mx + b\).
The reason it's so useful is because it transparently shows two crucial features of the line: the slope \( m \) and the y-intercept \( b \). The equation becomes easy to interpret once it's in this form.
The reason it's so useful is because it transparently shows two crucial features of the line: the slope \( m \) and the y-intercept \( b \). The equation becomes easy to interpret once it's in this form.
- \( m \) - the slope, which tells you how steep the line is.
- \( b \) - the y-intercept, where the line crosses the y-axis.
Plotting Points
Plotting points is an integral part of graphing linear equations, as it helps in pinpointing exact locations on a graph to draw the line.
To correctly graph a line, you must start with known points, usually beginning with the y-intercept.
To correctly graph a line, you must start with known points, usually beginning with the y-intercept.
- Start by plotting the y-intercept (0, -4) based on the slope-intercept equation \( y = 2x - 4 \).
- Use the slope to identify a second point: from (0, -4), move up 2 units and to the right by 1 unit, arriving at the point (1, -2).
Linear Equations
Linear equations represent straight lines on a graph and are foundational in algebra. They describe a constant rate of change, which is depicted by the slope.
For example, our equation \( y = 2x - 4 \) represents a line where any change in \( x \) by 1 unit leads to a change in \( y \) by 2 units. Linear equations are straight-forward because they only require a basic set of operations: addition, subtraction, multiplication, and division.
These operations maintain the equation's balance while allowing you to identify both the slope and intercept.
For example, our equation \( y = 2x - 4 \) represents a line where any change in \( x \) by 1 unit leads to a change in \( y \) by 2 units. Linear equations are straight-forward because they only require a basic set of operations: addition, subtraction, multiplication, and division.
These operations maintain the equation's balance while allowing you to identify both the slope and intercept.
Slope and Y-Intercept
The slope and y-intercept are key components of any linear equation, determining the line's direction and starting point.Slope:
- The slope is the term \( m \) in the equation \( y = mx + b \).
- It dictates the line's steepness and direction: positive slopes rise to the right, while negative ones fall.
- Our slope \( m \) is 2, indicating the line climbs 2 units up for every additional unit in the x-axis.
- The y-intercept \( b \) is where the line crosses the y-axis, at point (0, \( b \)).
- In this case, \( b \) is -4, so the line crosses the y-axis at -4.
