Problem 34
Question
Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Slope \(\frac{2}{3}\) and \(y\) -intercept -8
Step-by-Step Solution
Verified Answer
The equation of the line is \( y = \frac{2}{3}x - 8 \).
1Step 1: Identify the Slope and Y-Intercept
We are given that the slope of the line is \( \frac{2}{3} \) and the \( y \)-intercept is \(-8\). This allows us to identify the values for \( m \) and \( b \) in the slope-intercept equation \( y = mx + b \). Here, \( m = \frac{2}{3} \) and \( b = -8 \).
2Step 2: Substitute in the Slope-Intercept Formula
Using the values identified for \( m \) and \( b \), substitute them into the formula \( y = mx + b \). This gives us the equation of the line: \( y = \frac{2}{3}x - 8 \).
Key Concepts
Slope-Intercept FormSlopeY-Intercept
Slope-Intercept Form
When learning about linear equations, one of the most important forms to know is the slope-intercept form. This form is expressed as \( y = mx + b \), where \( y \) represents the dependent variable and \( x \) the independent variable. The other components of this form are \( m \), which denotes the slope of the line, and \( b \), the y-intercept.
This form is widely used because of its simplicity and ability to provide clear insights into a linear function's characteristics. By looking at the equation, you can easily identify the slope and y-intercept of the line. When plotted, this line will show how \( y \) changes with respect to \( x \).
Understanding this form allows you to quickly interpret and graph linear relationships, making problem-solving more straightforward.
This form is widely used because of its simplicity and ability to provide clear insights into a linear function's characteristics. By looking at the equation, you can easily identify the slope and y-intercept of the line. When plotted, this line will show how \( y \) changes with respect to \( x \).
- The slope \( m \) tells us how steep the line is.
- The y-intercept \( b \) indicates where the line crosses the y-axis.
Understanding this form allows you to quickly interpret and graph linear relationships, making problem-solving more straightforward.
Slope
Slope is a crucial concept in understanding the direction and steepness of a line. In the equation of a line represented by slope-intercept form \( y = mx + b \), the slope is denoted by \( m \).
The slope quantifies how much the dependent variable \( y \) changes for a unit change in the independent variable \( x \). A positive slope indicates that the line is increasing, while a negative slope shows it is decreasing. A larger absolute value of the slope means a steeper line, whereas a smaller or zero value means a flatter line.
For our original problem, the slope \( m = \frac{2}{3} \) means for every 3 units of horizontal change, the line will rise by 2 units, illustrating its moderate incline.
The slope quantifies how much the dependent variable \( y \) changes for a unit change in the independent variable \( x \). A positive slope indicates that the line is increasing, while a negative slope shows it is decreasing. A larger absolute value of the slope means a steeper line, whereas a smaller or zero value means a flatter line.
- Positive slope: Line rises from left to right.
- Negative slope: Line falls from left to right.
- Zero slope: Line is horizontal.
- Undefined slope: Line is vertical, but this is not typically expressed in slope-intercept form.
For our original problem, the slope \( m = \frac{2}{3} \) means for every 3 units of horizontal change, the line will rise by 2 units, illustrating its moderate incline.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis in a graph, represented by \( b \) in the slope-intercept form \( y = mx + b \).
This intercept is particularly important because it provides the starting point of the graph when \( x = 0 \). Essentially, it tells us the value of \( y \) when there is no influence from the variable \( x \).
For the problem at hand, the y-intercept is \(-8\). This implies that when the line is graphed, it will cross the y-axis at the point \( (0, -8) \).
Knowing the y-intercept helps in quickly sketching the graph and understanding the function's baseline value on the y-axis. It's a fundamental part of the linear equation that gives you a visual starting reference for how the line progresses.
This intercept is particularly important because it provides the starting point of the graph when \( x = 0 \). Essentially, it tells us the value of \( y \) when there is no influence from the variable \( x \).
For the problem at hand, the y-intercept is \(-8\). This implies that when the line is graphed, it will cross the y-axis at the point \( (0, -8) \).
Knowing the y-intercept helps in quickly sketching the graph and understanding the function's baseline value on the y-axis. It's a fundamental part of the linear equation that gives you a visual starting reference for how the line progresses.
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Problem 34
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