Problem 62
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. $$4 x+6 y+12=0$$
Step-by-Step Solution
Verified Answer
The slope-intercept form of the given equation is \(y = -2/3x - 2\), with slope -2/3 and y-intercept -2.
1Step 1: Rewrite in slope-intercept form
The given equation is \(4x + 6y + 12 = 0\). We can rearrange this equation to solve for y: \[\[-(4x) - 12\] on both sides of the equation to isolate 'y'. This gives us \(y = \[-(4/6)x -12/6\], simplifying further gives us \(y = -2/3x - 2\]. So, the equation in slope-intercept form is \(y = -2/3x - 2\).
2Step 2: Identifying Slope and Y-Intercept
In the slope-intercept form of the linear equation \(y = mx + b\), 'm' is the slope and 'b' is the y-intercept. Thus, in the rewritten equation \(y = -2/3x - 2\), the slope 'm' is -2/3 and the y-intercept 'b' is -2.
3Step 3: Graphing the Linear Function
The y-intercept is -2. This is where the line crosses the y-axis. So, start by plotting a dot at that point (-2) on the y-axis. The slope of the line is -2/3, which means for each positive step along the x-axis, we go two steps down, and for each negative step along the x-axis, we go two steps up. Draw the line passing through these points.
Key Concepts
Slope-Intercept FormLinear FunctionY-intercept
Slope-Intercept Form
The slope-intercept form is an equation of a straight line and is perhaps the most convenient form for graphing linear equations. It is written as
To convert a linear equation to slope-intercept form, you need to solve for
y = mx + b, where m represents the slope of the line and b represents the y-intercept, which is the point where the line crosses the y-axis.To convert a linear equation to slope-intercept form, you need to solve for
y so that it is isolated on one side of the equation. For the given problem, starting from 4x + 6y + 12 = 0, we subtract 4x and 12 from both sides to get 6y = -4x - 12. Next, we divide each term by 6 to solve for y, resulting in y = -2/3x - 2, which is now in slope-intercept form.Linear Function
A linear function is a function whose graph is a straight line. Linear functions are described by linear equations, which relate two variables in such a way that if plotted on a Cartesian plane, they would form a line. These equations can always be written in the form of
For instance, in our exercise
y = mx + b, where each variable and coefficient represents a fundamental characteristic of the line.For instance, in our exercise
y = -2/3x - 2, the linear function indicates a constant rate of change of y with respect to x. This constant rate is the slope -2/3, which means that for every unit increase in x, y decreases by 2/3 units, demonstrating a consistent, linear relationship between the two variables.Y-intercept
The y-intercept is the point where the graph of a function or an equation crosses the y-axis. It is a significant feature in graphing because it provides a starting point for drawing the graph of the equation. The y-intercept is represented by the
In the provided problem, the y-intercept is
b in the slope-intercept equation y = mx + b.In the provided problem, the y-intercept is
-2. This means that when x is 0, y is -2. When graphing, you would begin by placing a point at (0, -2) on the graph. You would then use the slope to determine the direction and steepness of the line, ensuring your graph accurately represents the linear relationship depicted by the equation.Other exercises in this chapter
Problem 62
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