Problem 21
Question
Use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) Point (0,-9) Slope \(m=-2\)
Step-by-Step Solution
Verified Answer
The three additional points on the line are (1,-11), (2,-13), (3,-15)
1Step 1: Identify the Given Point and Slope
The point given is (0,-9) and the slope 'm' is -2.
2Step 2: Use Slope to Determine Additional Points
Since slope is the change in y for every change in x, for every increase in x by 1, y will decrease by 2 (slope is -2). So, starting from the point (0,-9): \n\n- If x=1: \(y = -9 - (2*1) = -11\), so the new point is (1,-11) \n- If x=2: \(y = -9 - (2*2) = -13\), so the new point is (2,-13) \n- If x=3: \(y = -9 - (2*3) = -15\), so the new point is (3,-15)
3Step 3: Check the Derived Points
All these derived points should follow the same pattern that when x is increased by 1, y should decrease by 2 (slope = -2). This check ensures that the derived points are correct and lie on the same line.
Key Concepts
Slope-Intercept FormLinear EquationsCoordinate Geometry
Slope-Intercept Form
Understanding the slope-intercept form of a linear equation is a fundamental skill in algebra. It allows us to write equations of straight lines and easily identify the slope and the y-intercept, which is where the line crosses the y-axis. The slope-intercept form is expressed as:
\[ y = mx + b \]
Here, \( m \) represents the slope of the line, which measures the steepness and direction. It's a ratio that describes how much the y-value changes for a unit change in the x-value. The \( b \) symbolizes the y-intercept, which is the value of \( y \) when \( x = 0 \).
For instance, in the original exercise, the equation of a line with a slope of -2 and a y-intercept of -9 can be written as:
\[ y = -2x - 9 \]
With this form, if you have either the slope or a point on the line (or both), you can determine various points that lie on the same line. It simplifies plotting the line on a graph or finding its intersection with other lines.
\[ y = mx + b \]
Here, \( m \) represents the slope of the line, which measures the steepness and direction. It's a ratio that describes how much the y-value changes for a unit change in the x-value. The \( b \) symbolizes the y-intercept, which is the value of \( y \) when \( x = 0 \).
For instance, in the original exercise, the equation of a line with a slope of -2 and a y-intercept of -9 can be written as:
\[ y = -2x - 9 \]
With this form, if you have either the slope or a point on the line (or both), you can determine various points that lie on the same line. It simplifies plotting the line on a graph or finding its intersection with other lines.
Linear Equations
Linear equations form the basis for understanding relationships between two variables. They create straight lines when graphed on a coordinate plane and can be represented in various forms, including the slope-intercept form mentioned earlier. A key characteristic of linear equations is that they have a constant rate of change, which means the graph of these equations is a straight line.
A general linear equation in two variables, x and y, can be expressed as:
\[ ax + by = c \]
where \( a \), \( b \), and \( c \) are constants. You'll notice that if you solve for \( y \), you can convert this equation into the slope-intercept form. Linear equations can be manipulated using algebra to solve for one variable in terms of the other, to find the point of intersection of two lines, or even to model real-life situations.
When looking at the steps provided in the solution to find additional points on the line, they directly applied the concept of the slope (\( -2 \)) to find how \( y \) changes as \( x \) increase. This is the heart of using linear equations in coordinate geometry.
A general linear equation in two variables, x and y, can be expressed as:
\[ ax + by = c \]
where \( a \), \( b \), and \( c \) are constants. You'll notice that if you solve for \( y \), you can convert this equation into the slope-intercept form. Linear equations can be manipulated using algebra to solve for one variable in terms of the other, to find the point of intersection of two lines, or even to model real-life situations.
When looking at the steps provided in the solution to find additional points on the line, they directly applied the concept of the slope (\( -2 \)) to find how \( y \) changes as \( x \) increase. This is the heart of using linear equations in coordinate geometry.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is where algebra meets geometry. It provides a link between algebraic equations and geometric curves by using a coordinate system. This system typically consists of two perpendicular lines called axes: the horizontal x-axis and the vertical y-axis.
Points are located using pairs of numbers called coordinates, where each number corresponds to a position on one of the axes. For example, the point (3, -15) from the solution indicates that it is located 3 units along the x-axis and 15 units down from the origin on the y-axis.
In the context of the exercise, we're interested in finding points along a straight line. With the initial point (0, -9) and the information that the slope is -2, you can predict how the y value will change as you move along the x-axis. This is how the additional points were found, illustrating that with just a slope and a single point, you can map out the entirety of a linear path on the coordinate plane.
Points are located using pairs of numbers called coordinates, where each number corresponds to a position on one of the axes. For example, the point (3, -15) from the solution indicates that it is located 3 units along the x-axis and 15 units down from the origin on the y-axis.
In the context of the exercise, we're interested in finding points along a straight line. With the initial point (0, -9) and the information that the slope is -2, you can predict how the y value will change as you move along the x-axis. This is how the additional points were found, illustrating that with just a slope and a single point, you can map out the entirety of a linear path on the coordinate plane.
Other exercises in this chapter
Problem 21
Determine whether the equation represents \(y\) as a function of \(x.\) $$y=\sqrt{x^{2}-1}$$
View solution Problem 21
Use the Vertical Line Test to determine whether y is a function of x. Describe how you can use a graphing utility to produce the given graph. $$0.25 x^{2}+y^{2}
View solution Problem 22
Evaluate the indicated function for \(f(x)=x^{2}-1\) and \(g(x)=x-2\) algebraically. If possible, use a graphing utility to verify your answer. $$(f+g)(1)$$
View solution Problem 22
Show that \(f\) and \(g\) are inverse functions algebraically. Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Describe the relation
View solution