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Magazine Chapter 3: Problem 35
Graph each linear equation. See Examples 6 through \(10 .\) $$ x+8 y=8 $$
Short Answer
Expert verified
Plot points (0,1), (8,0), and (4,0.5) then draw a line through them.
Step by step solution
01
Understanding the Equation
The given equation is \( x + 8y = 8 \). This is a linear equation in two variables, which represents a straight line when graphed on the coordinate plane. Our task is to find points on this line and plot them.
02
Rearrange the Equation to Slope-Intercept Form
The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Rearranging our equation, we express \( y \) in terms of \( x \):\[8y = -x + 8 \y = -\frac{1}{8}x + 1\]This shows the slope \( m = -\frac{1}{8} \) and the y-intercept \( b = 1 \).
03
Identify Intercepts
To graph the line, find the intercepts where it crosses the x-axis and y-axis.- **Y-intercept:** Set \( x = 0 \): \[y = 1 \] So, the y-intercept is \((0, 1)\).- **X-intercept:** Set \( y = 0 \): \[x + 8(0) = 8 \x = 8 \] So, the x-intercept is \((8, 0)\).
04
Plotting Intercepts and Drawing the Line
On the coordinate plane, plot the points from the calculated intercepts: \((0, 1)\) and \((8, 0)\). Using a ruler, draw a straight line passing through these two points. This line represents the equation \( x + 8y = 8 \).
05
Verify with a Third Point
To ensure accuracy, determine another point on the line by choosing a value for \( x \).Set \( x = 4 \), then: \[4 + 8y = 8 \8y = 4 \y = \frac{1}{2} \] So, the point \((4, \frac{1}{2})\) should also lie on the line. Plot this point to confirm the correctness of the graph.
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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 is a popular way of expressing a linear equation. It follows the format \( y = mx + b \). Here, \(m\) represents the slope of the line, while \(b\) is the y-intercept. This form is very useful because it makes graphing linear equations simple and straightforward.
By rearranging equations into slope-intercept form, we can easily identify these features and start plotting the line on a graph.
- The **slope** \((m)\) tells us how steep the line is. It's the ratio of the vertical change to the horizontal change between two points on the line.
- The **y-intercept** \((b)\) indicates where the line crosses the y-axis. This is the point where the value of \(x\) is zero.
By rearranging equations into slope-intercept form, we can easily identify these features and start plotting the line on a graph.
X-intercept
The x-intercept of a line is the point where it crosses the x-axis. At this location, the value of \(y\) is always zero, which simplifies calculations. To find the x-intercept:
For the equation given in the exercise, \(x + 8y = 8\), substituting \(y = 0\) gives us:\[x + 8(0) = 8 \x = 8\]This means the x-intercept is at the point \((8, 0)\). Identifying the x-intercept helps us when plotting the line, providing a clear starting point on the x-axis.
- Substitute \(y = 0\) into the equation.
- Solve for \(x\).
For the equation given in the exercise, \(x + 8y = 8\), substituting \(y = 0\) gives us:\[x + 8(0) = 8 \x = 8\]This means the x-intercept is at the point \((8, 0)\). Identifying the x-intercept helps us when plotting the line, providing a clear starting point on the x-axis.
Y-intercept
The y-intercept shows where a line intersects the y-axis. At this point, \(x = 0\), and you can find the y-intercept by setting \(x\) to zero in the equation. This results in:
In our example equation \(x + 8y = 8\), plugging in \(x = 0\) results in:\[8y = 8 \y = 1\]Therefore, the y-intercept is \((0, 1)\).
- Set \(x = 0\) and solve for \(y\).
In our example equation \(x + 8y = 8\), plugging in \(x = 0\) results in:\[8y = 8 \y = 1\]Therefore, the y-intercept is \((0, 1)\).
- Remember, this is where the line crosses the y-axis, helping us identify one fixed point to start our graph.
Coordinate plane
The coordinate plane is a two-dimensional space where we graph equations. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis).
The coordinate plane is essential for visualizing algebraic equations, allowing us to see and confirm solutions. With intercepts like \((8, 0)\) and \((0, 1)\), you can draw and interpret the line equation with ease.
- **Coordinates**: Points on this plane are denoted as \((x, y)\), showing their exact position.
- **Quadrants**: The plane is divided into four sections or quadrants. Points in each quadrant have different signs for their coordinates.
- **Plotting lines**: To graph a line, such as \(x + 8y = 8\), plot the intercepts and draw a line through these points.
The coordinate plane is essential for visualizing algebraic equations, allowing us to see and confirm solutions. With intercepts like \((8, 0)\) and \((0, 1)\), you can draw and interpret the line equation with ease.
