Problem 59
Question
a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Graph the equation. $$3 x-9=0$$
Step-by-Step Solution
Verified Answer
The given equation in slope-intercept form is \(y = 0x + 3\). The slope of the line is 0 and the y-intercept is 3. The line is graphed as a horizontal line passing through the y-intercept at point (0, 3).
1Step 1: Rewrite in slope-intercept form
The slope-intercept form is also known as \(y = mx + c\), where m is the slope of the line and c is the y-intercept. For the given equation \(3x - 9 = 0\), it can be rewritten in slope intercept form as follows: Add 9 to both sides of the equation to isolate the term containing \(x\) to result into \(3x = 9\). Then, divide both sides by 3 to get \(x = 3\). Since there is no \(y\) in the equation, this can be written as \(y = 0x + 3\), emphasizing that the slope \(m = 0\) and the y-intercept \(c = 3\).
2Step 2: Identify the slope and y-intercept
In the equation \(y = 0x + 3\), the slope \(m = 0\) and the y-intercept \(c = 3\). The slope of zero signifies that the line is parallel to the x-axis, while the y-intercept indicates where the line crosses the y-axis, which is at point (0, 3).
3Step 3: Graph the equation
To graph the equation, plot the y-intercept point (0, 3) on the coordinate plane. Since the slope of the line is zero, this means that the line is horizontal and passes through the y-intercept. Draw a horizontal line through the y-intercept point to represent the equation.
Key Concepts
Graphing Linear EquationsSlope and Y-InterceptHorizontal LinesAlgebraic Equations
Graphing Linear Equations
Graphing linear equations is an essential skill in algebra. It involves plotting equations on a coordinate plane, which helps visualize the relationship between variables. One of the most common forms of linear equations is the slope-intercept form, given as \(y = mx + c\). Here, \(m\) is the slope and \(c\) is the y-intercept. Knowing how to graph these equations helps in understanding and predicting how changes in variables affect the outcome.
To graph a linear equation:
To graph a linear equation:
- Identify the slope and y-intercept from the equation.
- Plot the y-intercept on the y-axis.
- Use the slope to determine the rise over run from the y-intercept and plot additional points.
- Draw a straight line through the points.
Slope and Y-Intercept
The slope and y-intercept are the key features of a linear equation in its slope-intercept form. The slope, represented by \(m\), describes the steepness or incline of a line. It is calculated as the ratio of the rise (vertical change) to the run (horizontal change) between any two points on the line. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. A slope of zero, as seen in horizontal lines, indicates a flat or level line with no increase or decrease.
The y-intercept, \(c\), is where the line crosses the y-axis. It is the value of \(y\) when \(x = 0\). In the equation \(y = 0x + 3\), the y-intercept is at \(y = 3\). This tells us that no matter the value of \(x\), the line will always cross the y-axis at point \(3\). Understanding these components helps in effectively sketching and interpreting linear graphs.
The y-intercept, \(c\), is where the line crosses the y-axis. It is the value of \(y\) when \(x = 0\). In the equation \(y = 0x + 3\), the y-intercept is at \(y = 3\). This tells us that no matter the value of \(x\), the line will always cross the y-axis at point \(3\). Understanding these components helps in effectively sketching and interpreting linear graphs.
Horizontal Lines
Horizontal lines are a special type of linear equation where the slope is zero. In algebraic terms, this means the equation has the form \(y = c\), where \(c\) is a constant. For example, the line given by \(y = 3\) is horizontal.
This line runs parallel to the x-axis without any inclination, meaning it doesn't rise or fall as it extends across the plane. All points on a horizontal line have the same y-value, signifying a lack of vertical change despite changes in the x-value. Particularly, for the line \(x = 3\), it is a vertical line, but in terms of horizontal lines in context, \(y\) is consistent, showcasing stability at any selected x-value across the plane.
This line runs parallel to the x-axis without any inclination, meaning it doesn't rise or fall as it extends across the plane. All points on a horizontal line have the same y-value, signifying a lack of vertical change despite changes in the x-value. Particularly, for the line \(x = 3\), it is a vertical line, but in terms of horizontal lines in context, \(y\) is consistent, showcasing stability at any selected x-value across the plane.
Algebraic Equations
Algebraic equations like linear equations express the equality of two expressions. They form the foundation of algebraic manipulation and problem-solving. In linear algebra, you often encounter the slope-intercept form, but equations can come in different forms such as standard form (\(Ax + By = C\)) and point-slope form (\(y - y_1 = m(x - x_1)\)). Each form has its use depending on the information given or what needs to be determined.
Transforming equations from one form to another, like from standard to slope-intercept form, can make graphing and extracting meaningful data easier. For example, the original equation \(3x - 9 = 0\) was manipulated to highlight the slope and y-intercept clearly in slope-intercept form, \(y = 0x + 3\).
By understanding different methods to express linear relationships, students can solve, interpret, and visualize equations more flexibly, which is especially useful in real-world applications.
Transforming equations from one form to another, like from standard to slope-intercept form, can make graphing and extracting meaningful data easier. For example, the original equation \(3x - 9 = 0\) was manipulated to highlight the slope and y-intercept clearly in slope-intercept form, \(y = 0x + 3\).
By understanding different methods to express linear relationships, students can solve, interpret, and visualize equations more flexibly, which is especially useful in real-world applications.
Other exercises in this chapter
Problem 59
What must be done to a function's equation so that its graph is shifted vertically upward?
View solution Problem 59
Find the domain of each function. $$ H(r)=\frac{4}{r^{2}+11 r+24} $$
View solution Problem 59
Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function i
View solution Problem 59
Find the average rate of change of the function from \(x_{1}\) to \(x_{2}.\) $$f(x)=\sqrt{x} \text { from } x_{1}=4 \text { to } x_{2}=9$$
View solution