Problem 59
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. $$3 x+y-5=0$$
Step-by-Step Solution
Verified Answer
The slope of the given equation, once transformed to the slope-intercept form, is -3 and the y-intercept is 5. The graph of this linear function would be a straight line passing through the y-intercept (0,5) and sloping downwards with every unitary increase in X.
1Step 1: Rewrite the equation into slope-intercept form
To rewrite the given equation into slope-intercept form, we first need to isolate 'y'. This can be done by bringing all the terms involving 'x' to one side of the equation. The equation then becomes \(y = -3x + 5\).
2Step 2: Identification of Slope and Y-intercept
In the equation \(y = -3x + 5\), '-3' is the coefficient of 'x', hence represents the slope while '5' is the y-intercept.
3Step 3: Graph the Linear Function
Begin by drawing a coordinate system. Plot the y-intercept, which is the point (0, 5). The slope of the line is -3, which tells us that for every positive step in the x-direction, the line takes 3 negative steps in the y-direction. From the y-intercept, move to another point on the line by rising 0 unit and running 1 unit to the left (since the slope is negative). Repeat this process to determine several points, then draw a line passing through these points, and you've graphed the linear function.
Key Concepts
Slope-Intercept FormSlope of a LineY-Intercept
Slope-Intercept Form
The slope-intercept form is an equation of a straight line in which the variables 'x' and 'y' represent coordinates on a graph. It takes the form of \(y = mx + b\), where 'm' stands for the slope of the line, and 'b' represents the y-intercept. This form is particularly useful because it directly provides the necessary information for graphing. To convert any linear equation to the slope-intercept form, you need to solve the equation for 'y' so that it looks like the aforementioned form.
To illustrate, let's consider the equation from our exercise, \(3x + y - 5 = 0\). By rearranging the terms to isolate 'y', one obtains the slope-intercept form \(y = -3x + 5\). Here, we can immediately identify the slope and y-intercept, allowing for easy graphing of the linear function.
Always remember, converting to slope-intercept form simplifies the graphing process, as it gives a clearer visualization of how the line appears on the Cartesian plane. This form makes it evident where the line crosses the y-axis and the angle at which the line inclines or declines.
To illustrate, let's consider the equation from our exercise, \(3x + y - 5 = 0\). By rearranging the terms to isolate 'y', one obtains the slope-intercept form \(y = -3x + 5\). Here, we can immediately identify the slope and y-intercept, allowing for easy graphing of the linear function.
Always remember, converting to slope-intercept form simplifies the graphing process, as it gives a clearer visualization of how the line appears on the Cartesian plane. This form makes it evident where the line crosses the y-axis and the angle at which the line inclines or declines.
Slope of a Line
The slope of a line is a measure of its steepness or the degree of incline and is represented by the variable 'm' in the slope-intercept equation. It is calculated as the change in the 'y' values divided by the change in the 'x' values between two points on the line. The formula for determining the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{{y2 - y1}}{{x2 - x1}}\).
A positive slope means the line is rising from left to right, while a negative slope means it is falling. If the slope is zero, the line is horizontal, indicating no change in 'y' as 'x' increases; an undefined slope, where we get a zero divisor, indicates a vertical line, showing no change in 'x' as 'y' varies.
In our example, the slope is '-3' which tells us that for every unit we move to the right on the 'x'-axis, we move down three units on the 'y'-axis. Slope provides essential information about the direction and the angle at which the line moves across the Cartesian plane.
A positive slope means the line is rising from left to right, while a negative slope means it is falling. If the slope is zero, the line is horizontal, indicating no change in 'y' as 'x' increases; an undefined slope, where we get a zero divisor, indicates a vertical line, showing no change in 'x' as 'y' varies.
In our example, the slope is '-3' which tells us that for every unit we move to the right on the 'x'-axis, we move down three units on the 'y'-axis. Slope provides essential information about the direction and the angle at which the line moves across the Cartesian plane.
Y-Intercept
The y-intercept of a line refers to the point where the line crosses the y-axis on the graph. In the slope-intercept form \(y = mx + b\), the y-intercept is denoted by 'b'. It tells us the value of 'y' when 'x' is zero. The y-intercept is significant as it is a starting point for graphing a line and represents the value of the dependent variable when the independent variable is absent.
For the equation in our exercise, the y-intercept is '5', which means the graph of the line will cross the y-axis at the point (0, 5). To graph a line using the y-intercept, you first plot the y-intercept on the graph, then use the slope to determine the direction and steepness of the line as you plot additional points.
Understanding the y-intercept allows you to draw more accurate representations of linear functions and interpret the graphs effectively. It provides a clear-cut scenario of the line's position relative to the origin and helps to understand the initial value represented by the function.
For the equation in our exercise, the y-intercept is '5', which means the graph of the line will cross the y-axis at the point (0, 5). To graph a line using the y-intercept, you first plot the y-intercept on the graph, then use the slope to determine the direction and steepness of the line as you plot additional points.
Understanding the y-intercept allows you to draw more accurate representations of linear functions and interpret the graphs effectively. It provides a clear-cut scenario of the line's position relative to the origin and helps to understand the initial value represented by the function.
Other exercises in this chapter
Problem 59
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complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$ x^{2}+y^{2}-6 y-7=0 $$
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