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Chapter 1: Problem 62

Find an equation of the line that satisfies the given conditions. Through \(A(-1,4) ;\) slope \(\frac{2}{3}\)

Short Answer

Expert verified
Equation is \(y = \frac{2}{3}x + \frac{14}{3}\).

Step by step solution

01

Identify the Point-Slope Formula

To find the equation of the line, we will use the point-slope formula: \[ y - y_1 = m(x - x_1) \] where \(m\) is the slope of the line, and \((x_1, y_1)\) is a point on the line.
02

Substitute Given Values

Plug the given point \(A(-1, 4)\) and the slope \(\frac{2}{3}\) into the point-slope formula. \[ y - 4 = \frac{2}{3}(x + 1) \] This substitution sets up the equation in a form related to our specific line.
03

Simplify the Equation

Distribute the slope \(\frac{2}{3}\) through the expression \((x + 1)\): \[ y - 4 = \frac{2}{3}x + \frac{2}{3} \] Next, add 4 to both sides to solve for \(y\): \[ y = \frac{2}{3}x + \frac{2}{3} + 4 \] Combine like terms: \[ y = \frac{2}{3}x + \frac{14}{3} \] This is now in slope-intercept form, \(y = mx + b\), where \(b\) is the y-intercept.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are fundamental in mathematics, representing lines on a graph. These equations are typically in the form \(Ax + By = C\) or \(y = mx + b\), which is known as the slope-intercept form. They define a straight line when plotted on a coordinate plane.
Linear equations allow us to model relationships between variables, often depicting how one quantity changes in response to another.
  • Standard Form: The standard form of a linear equation is \(Ax + By = C\) where \(A\), \(B\), and \(C\) are constants, and \(x\) and \(y\) are variables.
  • Solution: Solving a linear equation means finding the value of \(x\) that fits the equation for a specific \(y\). A single linear equation in two variables, typically, has an infinite number of solutions represented by a line.
These are the basic building blocks of more complex algebra and calculus. Understanding them is essential for advancing in mathematical studies.
Slope-Intercept Form
The slope-intercept form of a linear equation, \(y = mx + b\), is a way of expressing the line's slope and intercept clearly. This form makes it easy to graph the equation.
  • Slope \(m\): The slope \(m\) of the line indicates its steepness and direction. A positive slope means the line ascends from left to right, while a negative slope means it descends. The slope is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
  • Y-intercept \(b\): This is the point where the line crosses the y-axis. It's the value of \(y\) when \(x = 0\). Knowing the y-intercept helps in graphically plotting the line quickly.
Using the slope-intercept form simplifies understanding of how changes in these values affect the appearance and position of the line on a graph. This form is especially helpful in identifying linear relationships in data analysis.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is the study of geometric figures using a coordinate system. This branch of geometry allows algebra to be used to solve geometric problems, providing a powerful tool for visualizing and solving equations.
  • Coordinate System: The most common system is the Cartesian coordinate system, which divides the plane into four quadrants using horizontal (x-axis) and vertical (y-axis) lines.
  • Points and Lines: Points on the plane are expressed as coordinates \((x, y)\). Lines, such as the one expressed in our exercise, can be represented using equations involving these coordinates.
  • Distance and Midpoint: Coordinate geometry also involves concepts like the distance between points and the midpoint of a segment, calculated using mathematical formulas.
Coordinate geometry is fundamental in linking algebra and geometry, offering an approach to solve real-world conditions through graphing and visualization.

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