Problem 24
Question
Write the slope-intercept equation of the line that passes through the two given points. $$ (2 \pi,-\pi),(0,0) $$
Step-by-Step Solution
Verified Answer
The slope-intercept equation is \( y = -\frac{1}{2}x \).
1Step 1: Identify the formula
To find the slope-intercept form of a line, use the formula: \[ y = mx + c \] where \( m \) is the slope and \( c \) is the y-intercept.
2Step 2: Find the slope
The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points \((2\pi, -\pi)\) and \((0, 0)\), the slope is \[ m = \frac{0 - (-\pi)}{0 - 2\pi} = \frac{\pi}{-2\pi} = -\frac{1}{2} \]
3Step 3: Find the y-intercept
To find the y-intercept \( c \), substitute \( m \), \( x \), and \( y \) from one of your points into the slope-intercept formula \( y = mx + c \). Using the point \((0,0)\), we see directly that \( c = 0 \) since: \[ 0 = -\frac{1}{2}(0) + c \] so \( c = 0 \).
4Step 4: Write the slope-intercept equation
With \( m = -\frac{1}{2} \) and \( c = 0 \), the equation of the line is: \[ y = -\frac{1}{2}x \]
Key Concepts
Linear EquationsSlope CalculationY-Intercept
Linear Equations
Linear equations are a foundational concept in algebra, representing lines on a graph. The general form of a linear equation is expressed as \( y = mx + c \), where "\( y \)" denotes the dependent variable and is usually plotted on the vertical axis of a graph.
On the other hand, "\( x \)" is the independent variable, plotted on the horizontal axis. In every linear equation, the coefficient \( m \) represents the slope of the line, while \( c \) signifies the y-intercept, which we'll discuss later in more depth.
Linear equations are important in understanding relationships between variables, as they provide a clear visualization of how changes in one quantity affect another. If you visualize this in a coordinate plane, a straight line connects every point that satisfies the equation.
Thus, knowing how to express an equation in its linear form is crucial because it makes it easier to predict and understand these relationships.
On the other hand, "\( x \)" is the independent variable, plotted on the horizontal axis. In every linear equation, the coefficient \( m \) represents the slope of the line, while \( c \) signifies the y-intercept, which we'll discuss later in more depth.
Linear equations are important in understanding relationships between variables, as they provide a clear visualization of how changes in one quantity affect another. If you visualize this in a coordinate plane, a straight line connects every point that satisfies the equation.
Thus, knowing how to express an equation in its linear form is crucial because it makes it easier to predict and understand these relationships.
Slope Calculation
The slope in a linear equation is a measurement that describes the rate of change between two variables. It's an indicator of how steep a line is. When calculating slope between two points on a graph, use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of the two points. The numerator \( (y_2 - y_1) \) represents the change in the \( y \)-values, while the denominator \( (x_2 - x_1) \) represents the change in the \( x \)-values.
Remember that a positive slope means the line rises from left to right, indicating a positive relationship between the variables. A negative slope, like in our solution with \( m = -\frac{1}{2} \), means the line descends, showing an inverse relationship.
Slopes can also be zero, creating a horizontal line, or undefined, creating a vertical line. Understanding the slope helps to interpret data and predict trends in the context of real-world scenarios.
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of the two points. The numerator \( (y_2 - y_1) \) represents the change in the \( y \)-values, while the denominator \( (x_2 - x_1) \) represents the change in the \( x \)-values.
Remember that a positive slope means the line rises from left to right, indicating a positive relationship between the variables. A negative slope, like in our solution with \( m = -\frac{1}{2} \), means the line descends, showing an inverse relationship.
Slopes can also be zero, creating a horizontal line, or undefined, creating a vertical line. Understanding the slope helps to interpret data and predict trends in the context of real-world scenarios.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept formula \( y = mx + c \), the \( c \) represents this intercept.
It signifies the value of \( y \) when \( x \) is zero. Physically, in terms of a graph, it's where the line attaches to the vertical line passing through \( x = 0 \).
Finding the y-intercept is straightforward when you have an equation like this. However, if you are dealing with points, you can determine the y-intercept by plugging in one of the points and the slope into the linear equation. In our example, using the point \((0,0)\), we found that \( c = 0 \), which means the line passes through the origin.
This aspect of linear functions is crucial because it gives you a point of reference, allowing you to easily sketch the line or understand its positioning within a given set of data.
It signifies the value of \( y \) when \( x \) is zero. Physically, in terms of a graph, it's where the line attaches to the vertical line passing through \( x = 0 \).
Finding the y-intercept is straightforward when you have an equation like this. However, if you are dealing with points, you can determine the y-intercept by plugging in one of the points and the slope into the linear equation. In our example, using the point \((0,0)\), we found that \( c = 0 \), which means the line passes through the origin.
This aspect of linear functions is crucial because it gives you a point of reference, allowing you to easily sketch the line or understand its positioning within a given set of data.
Other exercises in this chapter
Problem 24
Write the given polynomial as a product of irreducible polynomials of degree one or two. \(x^{3}+x^{2}-4 x-4\)
View solution Problem 24
State which of the six trigonometric functions are positive when evaluated at \(\theta\) in the indicated interval. \(\theta \in(3 \pi / 2,2 \pi)\)
View solution Problem 24
The Cartesian equation of a parabola is given. Determine its vertex and axis of symmetry. \(y=3 x^{2}-6 x+1\)
View solution Problem 24
Plot several points, and sketch the graph of the function defined by the given expression. $$ \left\\{\begin{aligned} x^{2}-4 & \text { if } x
View solution