Problem 36
Question
Determine the slope and y-intercept (if possible) of the linear equation. Then describe its graph. $$3 x+4 y=1$$
Step-by-Step Solution
Verified Answer
The slope of the line is -3/4 and the y-intercept is 1/4. The graph is a straight line which starts from point (0, 1/4) and goes downward as it moves to the right.
1Step 1: Rewrite the Equation in y = mx + c Form
First, we need to rewrite the given equation in the slope-intercept form. Let's manipulate the given equation \(3x + 4y = 1\). Subtract \(3x\) from both sides, which yields the equation: \(4y = -3x + 1\). Then, divide all terms by 4, the final equation becomes \(y = -3/4x + 1/4\)
2Step 2: Identify the Slope and Y-intercept
From the equation obtained in step 1, which is \(y = -3/4x + 1/4\), it's clear that the slope 'm' is -3/4 and the y-intercept 'c' is 1/4.
3Step 3: Describe the Graph
The graph of this equation is a straight line, because it's a linear equation. It starts from the y-intercept point (0, 1/4) on the y-axis. The slope of -3/4 means for each step right along the x-axis, the line moves 3/4 steps down. If slope were positive, it would move up.
Key Concepts
Slope-Intercept FormGraphing Linear EquationsSlope of a LineY-intercept of a Line
Slope-Intercept Form
Understanding the slope-intercept form is pivotal for graphing and analyzing linear equations. It expresses a linear equation in the form of \(y = mx + c\), where \(m\) stands for the slope of the line and \(c\) denotes the y-intercept, the point where the line crosses the y-axis.
As an example, let's look at the given exercise, which provides the equation \(3x + 4y = 1\). To convert this into slope-intercept form, we perform algebraic manipulations to solve for \(y\). By isolating \(y\) on one side, we determine that \(y = -\frac{3}{4}x + \frac{1}{4}\). This equation now clearly exhibits the slope \(m = -\frac{3}{4}\) and the y-intercept \(c = \frac{1}{4}\).
Whenever you encounter a linear equation, rewriting it in the slope-intercept form will immediately reveal the slope and y-intercept, making it easier to understand and graph the equation.
As an example, let's look at the given exercise, which provides the equation \(3x + 4y = 1\). To convert this into slope-intercept form, we perform algebraic manipulations to solve for \(y\). By isolating \(y\) on one side, we determine that \(y = -\frac{3}{4}x + \frac{1}{4}\). This equation now clearly exhibits the slope \(m = -\frac{3}{4}\) and the y-intercept \(c = \frac{1}{4}\).
Whenever you encounter a linear equation, rewriting it in the slope-intercept form will immediately reveal the slope and y-intercept, making it easier to understand and graph the equation.
Graphing Linear Equations
Plotting a line on a graph requires an understanding of the linear equation that represents it. With the slope-intercept form \(y = mx + c\), graphing becomes a two-step process. First, you plot the y-intercept \(c\) on the y-axis. Next, you use the slope \(m\) to determine the direction and steepness of the line.
The slope tells us how to move from the y-intercept to another point on the line. A positive slope indicates an upward trend from left to right, while a negative slope means the line goes downward as it moves to the right. For the given exercise, one would start at the y-intercept \((0, \frac{1}{4})\) and then follow the slope of \(-\frac{3}{4}\). This means for every 4 units you move to the right (positive direction on the x-axis), you must move 3 units down (negative direction on the y-axis). Connect these points and extend the line in both directions, and you've successfully graphed the linear equation.
The slope tells us how to move from the y-intercept to another point on the line. A positive slope indicates an upward trend from left to right, while a negative slope means the line goes downward as it moves to the right. For the given exercise, one would start at the y-intercept \((0, \frac{1}{4})\) and then follow the slope of \(-\frac{3}{4}\). This means for every 4 units you move to the right (positive direction on the x-axis), you must move 3 units down (negative direction on the y-axis). Connect these points and extend the line in both directions, and you've successfully graphed the linear equation.
Slope of a Line
The slope of a line is a measure of its steepness and direction. Mathematically, it's the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. It's represented by the variable \(m\) in the slope-intercept form of a linear equation, \(y = mx + c\).
For example, a slope of \(-\frac{3}{4}\) indicates that for every 4 units of horizontal movement (to the right if positive or to the left if negative), the vertical movement is 3 units down (because the slope is negative). If the slope were positive, the vertical movement would be upwards. It's essential to grasp that a steep line has a larger slope value, while a flatter line has a smaller slope value. Moreover, a horizontal line has a slope of 0, and a vertical line's slope is undefined.
For example, a slope of \(-\frac{3}{4}\) indicates that for every 4 units of horizontal movement (to the right if positive or to the left if negative), the vertical movement is 3 units down (because the slope is negative). If the slope were positive, the vertical movement would be upwards. It's essential to grasp that a steep line has a larger slope value, while a flatter line has a smaller slope value. Moreover, a horizontal line has a slope of 0, and a vertical line's slope is undefined.
Y-intercept of a Line
The y-intercept of a line is the point at which the line crosses the y-axis. It is an important concept in understanding linear equations because it helps to anchor the line on the graph. The y-intercept is represented by the variable \(c\) in the slope-intercept equation \(y = mx + c\), and it dictates where the line will start on the graph.
In our exercise, the y-intercept is \(\frac{1}{4}\), which means that the line crosses the y-axis at the point \((0, \frac{1}{4})\). To plot this, simply find the value on the y-axis and make a mark. It's important to note that every linear equation will have a y-intercept, but not all lines will have an x-intercept, especially those that are horizontal lines (unless they lie entirely on the x-axis). Identifying the y-intercept simplifies plotting the line and understanding its placement relative to the origin.
In our exercise, the y-intercept is \(\frac{1}{4}\), which means that the line crosses the y-axis at the point \((0, \frac{1}{4})\). To plot this, simply find the value on the y-axis and make a mark. It's important to note that every linear equation will have a y-intercept, but not all lines will have an x-intercept, especially those that are horizontal lines (unless they lie entirely on the x-axis). Identifying the y-intercept simplifies plotting the line and understanding its placement relative to the origin.
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Problem 36
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