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 equation in slope-intercept form is \(y = -2/3x - 2\). The slope is -2/3 and the y-intercept is -2. Using these values, we can draw the graph of the equation.
1Step 1: Rewrite the equation in slope-intercept form
Firstly, rearrange the given equation 4x + 6y + 12 = 0 to obtain a form y = mx + b by isolating y. This is done as: \[6y = -4x -12 \]\[y = \frac{-4}{6}x - \frac{12}{6}\] So y = \(-\frac{2}{3}x - 2\).
2Step 2: Identify the slope and y-intercept
In the slope-intercept form of the equation, y = mx + b, the coefficient of x is the slope and the constant term is the y-intercept. Here, m = -2/3 is the slope and b = -2 is the y-intercept.
3Step 3: Graph the linear function
Start by plotting the y-intercept (-2) on the y-axis. The slope is -2/3, which means for every 3 units moved to right on the x-axis, we move 2 units down. Following this pattern, we can plot other points and draw the line.
Key Concepts
Linear Function GraphingSlope of a LineY-Intercept
Linear Function Graphing
Graphing a linear function involves translating the algebraic representation of a line into a visual format on a coordinate system. For instance, if we have an equation like \(4x + 6y + 12 = 0\), our first task is to transform it into slope-intercept form, which is \(y = mx + b\), where \(m\) represents the slope, and \(b\) represents the y-intercept. By doing so, we get \(y = -\frac{2}{3}x - 2\).
To graph this equation, we start by locating the y-intercept on the y-axis, which is the point at which the line crosses the y-axis. In this case, the y-intercept is \(b = -2\). From there, we use the slope, which tells us the steepness and direction of the line. The slope of \(m = -\frac{2}{3}\) guides us to move 3 units horizontally (to the right for positive and to the left for negative) and 2 units vertically (up for positive and down for negative) to find the next point. Repeating this process, we can plot several points and draw a straight line through them, completing the graph.
To graph this equation, we start by locating the y-intercept on the y-axis, which is the point at which the line crosses the y-axis. In this case, the y-intercept is \(b = -2\). From there, we use the slope, which tells us the steepness and direction of the line. The slope of \(m = -\frac{2}{3}\) guides us to move 3 units horizontally (to the right for positive and to the left for negative) and 2 units vertically (up for positive and down for negative) to find the next point. Repeating this process, we can plot several points and draw a straight line through them, completing the graph.
Slope of a Line
The slope is a measure of how steep a line is, and the direction in which it slants. The formula for slope (\(m\)) when given two points, \((x_1, y_1)\) and \((x_2, y_2)\), is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In the slope-intercept form of a linear equation, \(y = mx + b\), the slope is represented by \(m\). For the equation \(6y = -4x - 12\), once rearranged to slope-intercept form, \(y = -\frac{2}{3}x - 2\), we can identify the slope as \(-\frac{2}{3}\).
When graphing, the slope tells us how to move from one point on the line to another. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. In our example, the slope of \(-\frac{2}{3}\) denotes a downward slant from left to right. For every three units we move horizontally, the line moves down two units vertically, showing the 'rise over run' relationship that defines slope.
When graphing, the slope tells us how to move from one point on the line to another. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. In our example, the slope of \(-\frac{2}{3}\) denotes a downward slant from left to right. For every three units we move horizontally, the line moves down two units vertically, showing the 'rise over run' relationship that defines slope.
Y-Intercept
The y-intercept is where a line crosses the y-axis on a graph. It is an important feature because it provides us with a starting point for drawing the line. In the slope-intercept form of a linear equation, \(y = mx + b\), the y-intercept is represented by the constant term \(b\).
Returning to our example equation in slope-intercept form, \(y = -\frac{2}{3}x - 2\), the y-intercept is -2. This means that the line crosses the y-axis at the point (0, -2). When graphing, we always begin by plotting the y-intercept. From there, we use the slope to determine the direction and steepness of the line, which helps in plotting the next points. The y-intercept is especially useful because, regardless of the slope, we have a guaranteed point on the line that we can be certain of and use as a reference when drawing the graph.
Returning to our example equation in slope-intercept form, \(y = -\frac{2}{3}x - 2\), the y-intercept is -2. This means that the line crosses the y-axis at the point (0, -2). When graphing, we always begin by plotting the y-intercept. From there, we use the slope to determine the direction and steepness of the line, which helps in plotting the next points. The y-intercept is especially useful because, regardless of the slope, we have a guaranteed point on the line that we can be certain of and use as a reference when drawing the graph.
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Problem 62
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