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Magazine Chapter 1: Problem 18
Write the slope-intercept equation of the line determined by the given data. Slope \(-5, x\) -intercept 5
Short Answer
Expert verified
The slope-intercept equation is \(y = -5x + 25\).
Step by step solution
01
Understanding the Given Data
We are provided with the slope of the line and the x-intercept. The slope is given as \(-5\) and the x-intercept is \(5\). An x-intercept of \(5\) means the point \((5, 0)\) lies on the line.
02
Identifying the Components of the Equation
The slope-intercept form of a line's equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Our task is to find this equation based on the given slope and the x-intercept.
03
Using the X-intercept to Find Y-intercept
At the x-intercept point \((5, 0)\), substitute \(x = 5\) and \(y = 0\) in the equation. Using the formula \(y = mx + b\):\[ 0 = -5(5) + b \]\[ 0 = -25 + b \]Solving for \(b\), we get:\[ b = 25 \]
04
Writing the Equation
Now that we have both the slope \(m = -5\) and the y-intercept \(b = 25\), substitute them into the slope-intercept form:\[ y = -5x + 25 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Equations
In mathematics, a linear equation represents a straight line when graphed on a coordinate plane. These equations are pivotal in understanding various real-life and mathematical scenarios where relationships between variables remain constant.
A linear equation is usually written in the form of the slope-intercept equation: \( y = mx + b \). In this formula, \( m \) is the slope, showing how steep the line is, and \( b \) represents the y-intercept, where the line crosses the y-axis. Because the relationship is linear, it means only one value of \( y \) exists for each value of \( x \).
A linear equation is usually written in the form of the slope-intercept equation: \( y = mx + b \). In this formula, \( m \) is the slope, showing how steep the line is, and \( b \) represents the y-intercept, where the line crosses the y-axis. Because the relationship is linear, it means only one value of \( y \) exists for each value of \( x \).
- These equations can model economic situations, physics problems, and everyday activities where consistent rates or relationships occur.
- Understanding them lays the groundwork for more complex functions and calculus.
X-Intercept
The x-intercept of a line is where the line crosses the x-axis on a graph, meaning at that point, the y-value is zero. In the slope-intercept form equation \( y = mx + b \), you find this intercept by setting \( y = 0 \) and solving for \( x \).
For example, consider the line from our exercise. Given the x-intercept is 5, it tells us that the line passes through the point \( (5, 0) \). This is crucial because it gives us a point to work from when constructing equations for lines, especially if the y-intercept or some other detail is unknown.
For example, consider the line from our exercise. Given the x-intercept is 5, it tells us that the line passes through the point \( (5, 0) \). This is crucial because it gives us a point to work from when constructing equations for lines, especially if the y-intercept or some other detail is unknown.
- In applications, the x-intercept can represent the initial condition before any changes have occurred.
- Finding this point is often the first step in many graphing exercises.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), this is represented by \( b \). It denotes the value of \( y \) when \( x = 0 \). This intercept is vital for forming equations quickly and can provide insight into the starting value of a situation being modeled.
In the exercise, we found the y-intercept by using the known x-intercept point \((5, 0) \) to solve for \( b \). By substituting \( x = 5 \) and \( y = 0 \) into the equation, we determined that \( b = 25 \).
In the exercise, we found the y-intercept by using the known x-intercept point \((5, 0) \) to solve for \( b \). By substituting \( x = 5 \) and \( y = 0 \) into the equation, we determined that \( b = 25 \).
- The y-intercept allows for quick graphing of the line if the slope is known.
- It helps in visually assessing how a variable changes over time from its original value.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, connects algebra and geometry using a coordinate system. This method lets us explore and solve broad geometric problems algebraically.
By plotting points like \((x, y)\) on a plane, we can describe geometric shapes with equations. The slope-intercept form is a cornerstone of this field, translating the linear equation's algebraic expression into visual representation on the plane.
By plotting points like \((x, y)\) on a plane, we can describe geometric shapes with equations. The slope-intercept form is a cornerstone of this field, translating the linear equation's algebraic expression into visual representation on the plane.
- Understanding coordinate geometry aids in solving problems through visual aids such as lines, curves, and shapes on graphs.
- It allows us to analyze spatial relationships using algebraic formulas.
