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Chapter 5: Problem 44

Solve the given problems. Find the value of \(k\) such that the line through \((0, k)\) and \((-k,-1)\) has a slope of \(1 / 2\).

Short Answer

Expert verified
The value of \(k\) is \(-2\).

Step by step solution

01

Understand the Slope Formula

The formula for the slope \( m \) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Since the slope is given as \( \frac{1}{2} \), we can equate this to the formula of the slope for the line through points \((0, k)\) and \((-k, -1)\).
02

Apply the Slope Formula

Insert the coordinates into the slope formula, where \( (x_1, y_1) = (0, k) \) and \( (x_2, y_2) = (-k, -1) \): \[ \frac{-1 - k}{-k - 0} = \frac{1}{2} \] Simplify the expression: \[ \frac{-1 - k}{-k} = \frac{1}{2} \]
03

Simplify the Slope Equation

The equation can be simplified: \[ \frac{-1 - k}{-k} = \frac{1}{2} \] Multiply both sides by \(-k\) to clear the fraction: \[ -1 - k = \frac{1}{2}(-k) \] This simplifies to: \[ -1 - k = -\frac{k}{2} \]
04

Solve for \(k\)

To eliminate the negative signs and fractions, multiply the entire equation by \(-2\): \[ 2(1 + k) = k \] Simplify and solve this equation: \[ 2 + 2k = k \] \[ 2k - k = -2 \] \[ k = -2 \]

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Key Concepts

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

Slope Formula
The slope formula is a fundamental concept in calculus and coordinate geometry, used to determine the steepness of a line. It describes how the y-value of a line change as the x-value increases. The formula for the slope, represented as \( m \), between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
This formula calculates the ratio of the vertical change (\( y_2 - y_1 \)) to the horizontal change (\( x_2 - x_1 \)) between two points. A positive slope indicates that the line is rising as it moves from left to right, while a negative slope indicates it is falling.
  • If the slope is zero, the line is horizontal.
  • An undefined slope indicates a vertical line.
  • In our problem, the slope is given as \( \frac{1}{2} \), meaning for every 2 units moved horizontally, there is a 1 unit increase vertically.
Linear Equations
Linear equations are equations that create a straight line when plotted on a graph. They are fundamental in algebra and calculus. The general form of a linear equation in two variables is:
\[y = mx + b\]
Here, \( m \) is the slope, and \( b \) is the y-intercept, the point where the line crosses the y-axis. Linear equations can be derived from the slope formula by rearranging terms to fit this form, allowing us to easily visualize the relationship between variables.
  • In our exercise, we used points to determine the equation of the line by employing the slope formula.
  • Subsequently, we solved for \( k \) ensuring the slope between the points matches the given value, \( \frac{1}{2} \).
Linear equations are powerful tools for modeling real-world situations, allowing us to predict outcomes and analyze trends.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, combines algebra and geometry to understand shapes and their properties through a coordinate system. By using the Cartesian plane, we can describe points, lines, and curves algebraically.
In our case:
  • We identified points \((0, k)\) and \((-k, -1)\) on the Cartesian plane.
  • Using these coordinates alongside the slope formula enabled us to express geometric relationships in precise algebraic terms.
Coordinate geometry allows us to bridge the visual and algebraic representations of mathematical concepts. This approach is crucial for solving problems where location and relationships between objects need to be demonstrated clearly. Understanding coordinates and slopes in geometry leads to deeper insights, like calculating distances and exploring more complex figures, reinforcing conceptual comprehension.

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Most popular questions from this chapter

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Solve the indicated or given systems of equations by an appropriate algebraic method. Solve the following system of equations by (a) solving the first equation for \(x\) and substituting, and (b) solving the first equation for \(y\) and substituting. $$\begin{aligned} &2 x+y=4\\\ &3 x-4 y=-5 \end{aligned}$$
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