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Magazine Chapter 3: Problem 34
Graph each linear equation. See Examples 2 through 6. $$ 2 x+y=2 $$
Short Answer
Expert verified
The graph is a line with slope -2 and y-intercept 2.
Step by step solution
01
Convert Equation to Slope-Intercept Form
Start by rearranging the given linear equation to the slope-intercept form, which is \( y = mx + b \). The given equation is \( 2x + y = 2 \). To convert it, solve for \( y \) by isolating it on one side of the equation. Subtract \( 2x \) from both sides: \( y = -2x + 2 \).
02
Identify the Slope and Y-Intercept
From the equation \( y = -2x + 2 \), identify the slope \( m \) and the y-intercept \( b \). The slope \( m \) is -2, and the y-intercept \( b \) is 2.
03
Plot the Y-Intercept on the Graph
The y-intercept \((0, 2)\) is a point where the line crosses the y-axis. Plot this point on the graph.
04
Use the Slope to Determine Another Point
The slope of -2 means that for every 1 unit you move to the right along the x-axis, you move 2 units down along the y-axis. From the y-intercept \((0, 2)\), move right 1 unit to \(x = 1\) and down 2 units to \(y = 0\). This gives another point, \((1, 0)\), to plot.
05
Draw the Line Through the Points
Using the points \((0, 2)\) and \((1, 0)\), draw a straight line through them to represent the equation \( y = -2x + 2 \). This is the graph of the given 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
The slope-intercept form of a linear equation is a straightforward way to graph a line. This form is given by the equation \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept. Converting an equation into this form makes it easier to see important properties of the line visually.
- The **slope** \( m \) describes the steepness and direction of the line. A positive slope indicates the line rises as you move from left to right, while a negative slope means the line falls.
- The **y-intercept** \( b \) is the point where the line crosses the y-axis. This is simply the value of \( y \) when \( x = 0 \).
Linear Equations
Linear equations are mathematical expressions that create straight lines when plotted on a graph. They are among the most fundamental and commonly used types of equations in algebra. Each linear equation can be expressed in various forms, including the slope-intercept form or the standard form.
- A linear equation often looks like \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.
- Solving a linear equation means finding all the pairs \((x, y)\) that satisfy the equation.
Plotting Points
Plotting points is a fundamental skill for graphing linear equations. This process involves placing points on a graph to visually represent solutions to the equation. Here's how you do it effectively:
- Start with the **y-intercept**, the easier point because its value on the y-axis is given directly鈥攍ike \((0, 2)\) in our example.
- Use the **slope** to find additional points. For a slope of \(-2\), from \( (0, 2) \), move 1 unit right to \(x = 1\) and 2 units down to \(y = 0\).
