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Magazine Chapter 3: Problem 22
Graph each linear equation. See Examples 2 through 6. $$ -x+5 y=5 $$
Short Answer
Expert verified
The graph is a line with slope \(\frac{1}{5}\) and y-intercept 1.
Step by step solution
01
Write the Equation in Slope-Intercept Form
Start by rewriting the given equation in the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This involves solving for \(y\). Begin by adding \(x\) to both sides of the equation to get:\[5y = x + 5\]
02
Isolate y
Next, divide every term by 5 to completely isolate \(y\):\[y = \frac{1}{5}x + 1\]This gives you the slope-intercept form: \(y = \frac{1}{5}x + 1\), where the slope \(m = \frac{1}{5}\) and the y-intercept \(b = 1\).
03
Plot the Y-Intercept
On a graph, locate the y-intercept, which is the point where the line crosses the y-axis. For \(y = \frac{1}{5}x + 1\), the y-intercept is \(b = 1\). Plot the point \((0, 1)\) on the graph.
04
Use the Slope to Find Another Point
The slope \(\frac{1}{5}\) means that for every unit you move horizontally to the right, you move \(\frac{1}{5}\) units vertically up. Starting at the y-intercept (\(0, 1\)), move 5 units to the right (to \(x = 5\)) and 1 unit up (to \(y = 2\)). Plot this second point \((5, 2)\).
05
Draw the Line
Once you have two points \((0, 1)\) and \((5, 2)\), draw a line through them. Extend this line across the graph, adding arrows on both ends to indicate that it continues indefinitely.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Linear Equations
Graphing linear equations involves creating a visual representation of an equation on a coordinate plane. This allows us to see the relationship between variables and understand how they interact. A linear equation is any equation that results in a straight line when graphed. To begin graphing a linear equation, ensure you place the equation in the correct format. By using accurate steps to identify necessary points, graphing becomes a straightforward task.
The key to graphing is identifying two or more specific points the line passes through, then connecting these points with a straight line.
The key to graphing is identifying two or more specific points the line passes through, then connecting these points with a straight line.
Slope-Intercept Form
The slope-intercept form of a linear equation is essential for graphing, as it simplifies finding the slope and y-intercept. This form is defined as:
- \(y = mx + b\)
- \(m\) represents the slope of the line.
- \(b\) represents the y-intercept, where the line crosses the y-axis.
Plotting Points
Plotting points accurately is a foundational skill in graphing linear equations. Once the equation is transformed into the slope-intercept form, the next step is to plot the y-intercept on the graph. This serves as your starting point.
After plotting the y-intercept, use the slope to find another point. Remember that the slope \(\frac{rise}{run}\) means the number of units to move vertically (rise) for each horizontal move (run). For instance:
After plotting the y-intercept, use the slope to find another point. Remember that the slope \(\frac{rise}{run}\) means the number of units to move vertically (rise) for each horizontal move (run). For instance:
- If the slope is \(\frac{1}{5}\), you move 1 unit up for every 5 units to the right.
Y-Intercept
The y-intercept is the point at which a line crosses the y-axis, and it is crucial in graphing linear equations. This point is easily identified in an equation written in slope-intercept form as the constant \(b\). For example:
- In the equation \(y = \frac{1}{5}x + 1\), the y-intercept is 1.
