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Magazine Chapter 3: Problem 42
Graph each linear equation. Plot four points for each line. $$x+2 y=4$$
Short Answer
Expert verified
Plot points: (-2, 3), (0, 2), (2, 1), (4, 0), and draw a line through them.
Step by step solution
01
Isolate y
Start by isolating the variable y in the equation. The given equation is \(x + 2y = 4\). Subtract x from both sides to get \(2y = 4 - x\). Then divide both sides by 2: \(y = 2 - \frac{x}{2}\). Now, the equation is in slope-intercept form: \(y = -\frac{1}{2}x + 2\).
02
Choose x-values
Select four different x-values to plot the points. Common choices are -2, 0, 2 and 4.
03
Calculate corresponding y-values
Substitute the chosen x-values into the equation \(y = -\frac{1}{2}x + 2\) to find the corresponding y-values: - For \(x = -2\): \(y = 2 - \frac{-2}{2} = 3\). - For \(x = 0\): \(y = 2 - \frac{0}{2} = 2\). - For \(x = 2\): \(y = 2 - \frac{2}{2} = 1\). - For \(x = 4\): \(y = 2 - \frac{4}{2} = 0\).
04
Plot the points
Plot the points you calculated on a coordinate grid: (-2, 3), (0, 2), (2, 1) and (4, 0).
05
Draw the line
Draw a straight line through all four points to represent the equation \(x + 2y = 4\). Extend the line across the entire graph for better visualization of the linear relationship.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Equations
Linear equations form the backbone of many algebraic concepts you will encounter. A linear equation is an equation in which the highest power of the variable is one. This means the graph of a linear equation will always be a straight line. Linear equations typically appear in the form of \(ax + by = c\). By manipulating this equation, you can find solutions that fit a line on a graph. This helps understand relationships and changes between variables.
Slope-Intercept Form
One useful form of a linear equation is the slope-intercept form, which is written as \(y = mx + b\). Here, \(m\) represents the slope of the line and \(b\) represents the y-intercept, or the point where the line crosses the y-axis.
This makes it easier to identify the slope and y-intercept, which will help in plotting the equation accurately.
- Slope (\(m\)): This describes how steep the line is. A positive slope means the line rises as it goes right, while a negative slope means it falls.
- Y-intercept (\(b\)): This is where the line touches the y-axis. It's the value you get for \(y\) when \(x\) is zero.
This makes it easier to identify the slope and y-intercept, which will help in plotting the equation accurately.
Coordinate Plane
A coordinate plane, also called a Cartesian plane, is a two-dimensional surface where we can graph equations. It is made up of two intersecting lines: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin, designated as \((0,0)\).
- X-axis: Runs left (negative) to right (positive).
- Y-axis: Runs down (negative) to up (positive).
Plotting Points
Plotting points on the coordinate plane involves marking specific coordinates obtained from an equation. For the equation \(x + 2y = 4\), we transform it to \(y = -\frac{1}{2}x + 2\) and then select values for \(x\) to find corresponding \(y\) values.
Follow these steps:
Follow these steps:
- Choose x-values: Common choices like -2, 0, 2, and 4.
- Calculate y-values: Use the slope-intercept equation. For example, when
\(x = -2\), then \(y = 2 - \frac{-2}{2} = 3\); when \(x = 0\), then \(y = 2\); when \(x = 2\), then \(y = 1\); and when \(x = 4\), then \(y = 0\). - Mark coordinates: Plot the points \((-2,3), (0,2), (2,1), (4,0)\) on the plane.
- Draw the line: Connect these points with a straight line extending through them.
