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Chapter 1: Problem 46

Write an equation of the line that passes through the points. Then use the equation to sketch the line. $$ \left(\frac{7}{8}, \frac{3}{4}\right),\left(\frac{5}{4},-\frac{1}{4}\right) $$

Short Answer

Expert verified
The equation of the line is \(y = -2x + \frac{17}{4}\)

Step by step solution

01

Finding the Slope

Calculate the slope of the line using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the coordinates of the given points, \(\left(\frac{7}{8}, \frac{3}{4}\right)\) and \(\left(\frac{5}{4},-\frac{1}{4}\right)\) results in \(m = \frac{-\frac{1}{4}-\frac{3}{4}}{\frac{5}{4}-\frac{7}{8}}\) which simplifies to \(m = -2\).
02

Writing the Equation of the Line

The equation of the line is written in the slope point form \(y - y_1 = m(x - x_1)\). Taking the slope as -2 and using the point \(\left(\frac{7}{8}, \frac{3}{4}\right)\), the equation becomes \(y - \frac{3}{4} = -2(x - \frac{7}{8})\). Multiplying through by 4 to clear the fractions gives the equation of the line as \(4y - 3 = -2(4x - 7)\) which simplifies to \(4y = -8x + 17\). Dividing through by 4 gives the final equation of the line as \(y = -2x + \frac{17}{4}\).
03

Sketching the Line

To sketch the line, plot the points \(\left(\frac{7}{8}, \frac{3}{4}\right)\) and \(\left(\frac{5}{4},-\frac{1}{4}\right)\) on the graph. Draw a straight line passing through these points, this line represents the equation \(y = -2x + \frac{17}{4}\).

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Key Concepts

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

Slope Calculation
The slope of a line is crucial in understanding how steep a line is or how it visually inclines and declines on a graph. Calculating the slope can be thought of as finding the 'tilt' of the line. To determine this, you will typically see the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( m \) represents the slope, and the coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points that lie on the line.
  • The numerator, \( y_2 - y_1 \), shows the change in the vertical direction (up or down movement on a graph).
  • The denominator, \( x_2 - x_1 \), indicates the change in the horizontal direction (left or right movement). When you divide these differences, the result is your slope.
  • For example, from the given points \( (\frac{7}{8}, \frac{3}{4}) \) and \( (\frac{5}{4}, -\frac{1}{4}) \), the slope calculated is \( -2 \).
This negative slope means the line decreases as it moves from left to right. Different types of slopes illustrate different scenarios:
  • A positive slope means rising from left to right.
  • A zero slope indicates a horizontal line.
  • A negative slope, like our example, shows a descending line going from left to right.
  • If the slope is undefined, it indicates a vertical line.
Point-Slope Form
Once the slope is known, creating a manageable equation to represent the line becomes the next task. For this, the point-slope form of the linear equation is particularly effective \( y - y_1 = m(x - x_1) \). This formula uses a known point from the line and the calculated slope to formulate the equation of the line.
  • \( m \) represents the slope that you've determined.
  • \( (x_1, y_1) \) is a point on the line. In our case, \( (\frac{7}{8}, \frac{3}{4}) \) is selected.
Substituting these values into the point-slope form, the equation becomes \( y - \frac{3}{4} = -2(x - \frac{7}{8}) \). This signifies the structure before further simplification:
  • Multiply through by 4 to eliminate fractions, simplifying to \( 4y - 3 = -2(4x - 7) \).
  • Finally, simplify fully to achieve the slope-intercept form, resulting in \( y = -2x + \frac{17}{4} \).
This results in your typical linear equation, fully ready for plotting onto a graph to visually see the described line.
Graphing Lines
Graphing a line is the visual representation of the linear equation on a coordinate plane. It helps in understanding the equation better by showing a visual interpretation. To begin graphing:
  • Start by plotting key points on the graph, like \( (\frac{7}{8}, \frac{3}{4}) \) and \( (\frac{5}{4}, -\frac{1}{4}) \).
  • Use a ruler to draw a straight line that passes through both points, embodying the behavior indicated by the equation \( y = -2x + \frac{17}{4} \).
It's important to remember that
  • The y-intercept of this line, \( \frac{17}{4} \), tells you where the line crosses the y-axis (vertical axis).
  • The slope \( -2 \) continues to describe the direction and steepness of the line as referenced through the line's steep tilt downwards.
As a result, you can visually follow the line moving from one plotted point to the other. This graph serves as a practical cross-check alongside the algebraic solution to ensure the line's accuracy and consistency. Offering a two-dimensional perspective, graphing confirms that the work on paper translates accurately into the space of a coordinate system.

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