Chapter 5: Problem 2
Graph the equations. \(x+2 y=2\)
Short Answer
Expert verified
Plot (0,1) and (2,0), then draw a line through these points.
Step by step solution
01
- Identify the Equation's Form
The given equation is in the form of a linear equation: \(x + 2y = 2\). This can be rewritten in slope-intercept form \(y = mx + b\) to easily graph it.
02
- Rewrite in Slope-Intercept Form
Start by solving for \(y\). Subtract \(x\) from both sides:\[2y = -x + 2\]Then divide by 2:\[y = -\frac{1}{2}x + 1\]Now the equation is in slope-intercept form \(y = mx + b\), where \(m = -\frac{1}{2}\) is the slope and \(b = 1\) is the y-intercept.
03
- Identify the Y-Intercept
From the slope-intercept form equation \(y = -\frac{1}{2}x + 1\), we see that the y-intercept \(b\) is 1. This means the line crosses the y-axis at the point (0, 1).
04
- Use the Slope to Find Another Point
The slope \(-\frac{1}{2}\) means that for every 1 unit increase in \(x\), \(y\) decreases by \(\frac{1}{2}\) unit. Starting from the y-intercept (0, 1), if we move 2 units to the right (\(x\) increases by 2), then \(y\) will decrease by 1:Starting from (0, 1): New point is (2, 0).
05
- Plot the Points and Draw the Line
Plot the y-intercept (0, 1) and the point (2,0) on a graph. Draw a straight line through these points to represent the equation \(x + 2y = 2\).
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
linear equations
Linear equations are fundamental in algebra and are a type of equation where the highest power of the variable is 1. They can be written in various forms, but the most common ones are the standard form, ax + by = c, and the slope-intercept form, y = mx + b. Linear equations graph as straight lines on a coordinate plane. The main components of a linear equation include the coefficients (a, b, and c in standard form), the slope (m), and the y-intercept (b in slope-intercept form). Understanding these concepts will help you rearrange equations and graph them effectively.
Examples of linear equations include: x + 2y = 2 and y = -0.5x + 1.
slope-intercept form
The slope-intercept form is very useful for graphing linear equations. It's written as y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. This form makes it easy to identify the key features of the line. The slope 'm' shows how steep the line is, and the y-intercept 'b' indicates where the line crosses the y-axis. For instance, in the equation y = -0.5x + 1, the slope is -0.5, meaning the line goes down 0.5 units for every 1 unit it goes right. The y-intercept is 1, indicating the line crosses the y-axis at (0, 1).
- To convert a standard form equation to slope-intercept form, isolate 'y' on one side of the equation.
- This makes it easier to graph the equation because you can quickly identify slope and y-intercept.
graph plotting
Graph plotting involves plotting points on a coordinate plane to represent an equation. For linear equations, you'll need at least two points to plot a line. Start by finding the y-intercept, which is an easy point to identify from the slope-intercept form. Then use the slope to find another point. For example, with the equation y = -0.5x + 1, start at (0, 1). Then, using the slope -0.5, move 2 units to the right and 1 unit down to find another point (2,0).
- Use graph paper or a digital tool to accurately plot your points.
- Once you've plotted your points, draw a straight line through them to complete the graph.
Remember to label your axes and scale your graph appropriately.
y-intercept
The y-intercept of a linear equation is the point where the graph crosses the y-axis. This is represented by 'b' in the slope-intercept form y = mx + b. To find the y-intercept from an equation, set x to 0 and solve for y. For example, in y = -0.5x + 1, when x = 0, y = 1, so the y-intercept is 1.
The y-intercept provides an easy starting point for graphing because it's an exact point on the graph. Once you have the y-intercept, you can use the slope to find more points. Just remember, the coordinate of the y-intercept is always in the form (0, b). Additionally, the y-intercept helps in understanding the behavior of the line in real-world applications, like initial values in economic models.
slope calculation
The slope of a line indicates its steepness and direction. It's calculated as the ratio of the change in y (rise) to the change in x (run). In mathematical terms, slope (m) is defined as \(\frac{rise}{run}\). For a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\), the slope is calculated as \(\frac{y_2 - y_1}{x_2 - x_1}\).
For instance, in the equation y = -0.5x + 1, the slope is -0.5. This means that for every 1 unit increase in x, y decreases by 0.5 units. Understanding the slope helps in graphing the line accurately.
- Positive slopes mean the line ascends as you move right.
- Negative slopes mean the line descends as you move right.
- A slope of zero means the line is horizontal.
- An undefined slope means the line is vertical.
Being able to calculate the slope will help you analyze and graph any linear equation quickly.
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