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Magazine Chapter 2: Problem 49
Find the slope and y-intercept of the line, and draw its graph. $$ \frac{1}{2} x-1 y+1=0 $$
Short Answer
Expert verified
Slope = \(\frac{1}{2}\), Y-intercept = 1.
Step by step solution
01
Rearrange the Equation into 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. Start by rearranging the given equation \( \frac{1}{2}x - 1y + 1 = 0 \) into this form. Firstly, isolate \( y \) on one side:\[ \frac{1}{2}x + 1 = y \].
02
Identify the Slope and Y-Intercept
Now that the equation \( y = \frac{1}{2}x + 1 \) is in slope-intercept form, identify the slope \( m \) and the y-intercept \( b \). Here, the slope \( m = \frac{1}{2} \) and the y-intercept \( b = 1 \).
03
Plot the Y-Intercept on the Graph
Start drawing the graph by plotting the y-intercept point \( (0, 1) \) on the y-axis.
04
Use the Slope to Determine Another Point
From the y-intercept (0, 1), use the slope \( \frac{1}{2} \). This means for every 2 units you move to the right (increasing x), you move up 1 unit (increasing y). Thus, from \( (0, 1) \), go right 2 units and up 1 unit to plot another point at \( (2, 2) \).
05
Draw the Line Through the Points
Draw a straight line through the points \( (0, 1) \) and \( (2, 2) \) to represent the line defined by the equation \( \frac{1}{2}x - y + 1 = 0 \). This line is the graphical representation 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 a mathematical equation that forms a straight line when graphed on a coordinate plane. It's called "linear" because it represents a line. These equations typically appear in the form \( ax + by + c = 0 \), where \( a \), \( b \), and \( c \) are constants. In our example, the equation given is \( \frac{1}{2}x - y + 1 = 0 \), which is a standard form of a linear equation. To work with this equation easily, we often convert it into the slope-intercept form, which simplifies graphing and understanding its features.
Slope
The slope of a line is a measure that describes how steep the line is. It indicates the direction and the rate of change as you move along the line. The slope \( m \) is calculated as the rise over run between any two points on the line.
- If \( m > 0 \), the line rises from left to right, meaning it has an upward slope.
- If \( m < 0 \), the line falls from left to right, signifying a downward slope.
- If \( m = 0 \), the line is perfectly horizontal.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. This point can be found by setting \( x = 0 \) in the equation and solving for \( y \). It is represented by the coordinate \((0, b)\), where \( b \) is the y-intercept value.
- If the y-intercept is positive, the line crosses above the origin.
- If the y-intercept is negative, the line crosses below the origin.
Graphing Lines
Graphing lines involves plotting points and drawing a line through them. With a linear equation in slope-intercept form, start by identifying the y-intercept and then use the slope to determine another point.
- First, plot the y-intercept. For \( y = \frac{1}{2}x + 1 \), plot the point \((0, 1)\).
- Next, use the slope. With a slope of \( \frac{1}{2} \), from \((0, 1)\), move 2 units right and 1 up to plot \((2, 2)\).
- Draw a straight line through the points \((0, 1)\) and \((2, 2)\). This visualizes the equation \( \frac{1}{2}x - y + 1 = 0 \).
