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Magazine Chapter 2: Problem 33
Exer. 33-36: Find the slope-intercept form of the line that satisfles the given conditions. \(x\) -intercept \(4, \quad y\) -intercept \(-3\)
Short Answer
Expert verified
The slope-intercept form is \( y = \frac{3}{4}x - 3 \).
Step by step solution
01
Understand slope-intercept form
The slope-intercept form of a line is given by \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept.
02
Determine the points given
We have two intercepts: the \( x \)-intercept is 4, which means the point is \((4, 0)\), and the \( y \)-intercept is -3, which means the point is \((0, -3)\).
03
Calculate the slope
The slope \( m \) of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting into the formula gives \( m = \frac{-3 - 0}{0 - 4} = \frac{-3}{-4} = \frac{3}{4} \).
04
Substitute into slope-intercept form formula
Now that we know the slope \( m = \frac{3}{4} \) and the \( y \)-intercept \( b = -3 \), we can substitute these into the slope-intercept form equation: \( y = \frac{3}{4}x - 3 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope Calculation
To calculate the slope of a line, you need two distinct points on the line. The slope is a measure of how steep the line is, and it is represented by the variable \( m \). The formula for calculating the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In the context of the problem, you are given an \( x \)-intercept at \((4, 0)\) and a \( y \)-intercept at \((0, -3)\). Substituting these into the formula, you get:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In the context of the problem, you are given an \( x \)-intercept at \((4, 0)\) and a \( y \)-intercept at \((0, -3)\). Substituting these into the formula, you get:
- \( y_1 = 0 \) and \( y_2 = -3 \)
- \( x_1 = 4 \) and \( x_2 = 0 \)
- \( m = \frac{-3 - 0}{0 - 4} = \frac{-3}{-4} = \frac{3}{4} \)
x-intercept
The \( x \)-intercept of a line is the point where the line crosses the \( x \)-axis. At this point, the value of \( y \) is zero. In simpler terms, it's where the line "touches" the \( x \)-axis.
In our exercise problem, the \( x \)-intercept is given as 4. This means that at the point \((4, 0)\), the line intersects the \( x \)-axis. To better understand, if you plug the \( x \)-intercept into a linear equation, you will set \( y=0 \) and solve for \( x \).
In our exercise problem, the \( x \)-intercept is given as 4. This means that at the point \((4, 0)\), the line intersects the \( x \)-axis. To better understand, if you plug the \( x \)-intercept into a linear equation, you will set \( y=0 \) and solve for \( x \).
- Verify by substitution: \( y = \frac{3}{4}x - 3 \)
- Set \( y = 0 \) to find \( x \): \( 0 = \frac{3}{4}(4) - 3 \).
- This confirms \( x = 4 \) for \( y = 0 \).
y-intercept
The \( y \)-intercept is where the line crosses the \( y \)-axis. At this point, \( x \) is always zero. It is symbolized as \( b \) in the slope-intercept form equation \( y = mx + b \).
In your given problem, the \( y \)-intercept is \(-3\). This indicates that when \( x = 0 \), the line will intersect the \( y \)-axis at the point \((0, -3)\). You can see this is easily substituted into the equation to find specific \( y \) values.
In your given problem, the \( y \)-intercept is \(-3\). This indicates that when \( x = 0 \), the line will intersect the \( y \)-axis at the point \((0, -3)\). You can see this is easily substituted into the equation to find specific \( y \) values.
- Use the slope-intercept form: \( y = \frac{3}{4}(0) - 3 \)
- \( y = -3 \), confirming the \( y \)-intercept at \((0, -3)\)
Linear Equations
Linear equations represent straight lines in a two-dimensional space, and their most commonly used form is the slope-intercept form, which is \( y = mx + b \). Let's explain what each part of this formula means:
Linear equations are foundational in understanding algebra, and their simplicity makes them invaluable tools for modeling real-world situations where relationships between two quantities are constant.
- \( y \): The dependent variable, representing the output or value of the line for a given \( x \).
- \( m \): The slope, indicating the steepness or incline of the line.
- \( x \): The independent variable, representing the input or what is controlled in the equation.
- \( b \): The \( y \)-intercept where the line crosses the \( y \)-axis.
Linear equations are foundational in understanding algebra, and their simplicity makes them invaluable tools for modeling real-world situations where relationships between two quantities are constant.
