Problem 61
Question
a. Rewrite the given equation in slope-intercept form. b. Give the slope and \(y\) -intercept. c. Use the slope and y-intercept to graph the linear function. $$2 x+3 y-18=0$$
Step-by-Step Solution
Verified Answer
The equation in slope-intercept form is \(y = -\frac{2}{3}x + 6\), with a slope of \(-\frac{2}{3}\) and a y-intercept at 6. The graph of the function is a downward sloping line that crosses the y-axis at the point (0,6).
1Step 1: Rewrite the equation in slope-intercept form
Rearrange the given equation \(2x + 3y - 18 = 0\) to isolate \(y\). This can be achieved by first subtracting \(2x\) from both sides of the equation to get \(3y = -2x + 18\). Then divide every term by 3 to solve for \(y\), yielding \(y = -\frac{2}{3}x + 6\).
2Step 2: Identify the slope and y-intercept
From the equation in slope-intercept form (\(y = mx + c\)), we can identify the slope (m) and y-intercept (c). In this case, the slope \(m = -\frac{2}{3}\) and the y-intercept \(c = 6\).
3Step 3: Graph the linear function
Begin by plotting the y-intercept on the graph at the point (0,6). Then, from this point, use the slope to find the next point. Since the slope is \(-\frac{2}{3}\), this means for every three units moving to the right on the x-axis, move two units down on the y-axis. Repeat this process to plot a few more points. Finally, draw a line passing through these points to graph the linear function.
Key Concepts
Linear EquationsGraphing Linear FunctionsSlope and Y-Intercept
Linear Equations
Linear equations are fundamental in mathematics and are characterized by combinations of variables and constants which, when plotted, form a straight line. These equations are typically expressed in the form \(Ax + By + C = 0\), where \(A\), \(B\), and \(C\) are constants. This type of equation represents a straight line in two-dimensional space and provides a crucial foundation for algebra and geometry.
To work effectively with linear equations, one often needs to convert them into a more useful form, such as the slope-intercept form. This makes solving and graphing these equations much easier, especially when analyzing how different variables interact with each other. Understanding linear equations is essential for solving real-world problems that involve constant rates of change.
Working with linear equations requires basic algebraic skills, like rearranging equations and solving for variables. It's also important to grasp the concept of a slope and an intercept, as they offer insights into the nature of the graph and how the variables relate.
To work effectively with linear equations, one often needs to convert them into a more useful form, such as the slope-intercept form. This makes solving and graphing these equations much easier, especially when analyzing how different variables interact with each other. Understanding linear equations is essential for solving real-world problems that involve constant rates of change.
Working with linear equations requires basic algebraic skills, like rearranging equations and solving for variables. It's also important to grasp the concept of a slope and an intercept, as they offer insights into the nature of the graph and how the variables relate.
Graphing Linear Functions
Graphing linear functions involves plotting their equation on a coordinate plane, leading to a visual representation of the function's behavior. This process is heavily dependent on the equation being in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
Graphing has several steps:
This process helps to quickly comprehend a linear function's behavior, including direction and steepness, facilitating deeper analysis and prediction of real-world scenarios.
Graphing has several steps:
- Convert the equation to slope-intercept form if it's not already.
- Identify the y-intercept \((0, c)\) and plot it on the Cartesian plane.
- Utilize the slope \(m\). For a slope of \(-\frac{2}{3}\), for example, you move three units to the right and two units down, creating another point.
- Draw the line through these points for a complete graph.
This process helps to quickly comprehend a linear function's behavior, including direction and steepness, facilitating deeper analysis and prediction of real-world scenarios.
Slope and Y-Intercept
The slope and y-intercept are critical in understanding and graphing linear equations. The slope \(m\) of a line refers to its steepness and direction, dictating how the line ascends or descends across the plane. It can be computed by the formula \( m = \frac{\text{rise}}{\text{run}} \), which represents the change in \(y\) over the change in \(x\).
The y-intercept \(c\) is the point where the line intersects the y-axis, representing the initial value of \(y\) when \(x\) is zero. Together, the slope and y-intercept offer a complete image of the line's position and trajectory on a graph.
The y-intercept \(c\) is the point where the line intersects the y-axis, representing the initial value of \(y\) when \(x\) is zero. Together, the slope and y-intercept offer a complete image of the line's position and trajectory on a graph.
- A positive slope means the line ascends from left to right.
- A negative slope indicates the line descends from left to right.
- If the slope is zero, the line is horizontal.
- The y-intercept tells where the line crosses the y-axis, easily found as the point \( (0, c) \).
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Problem 61
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