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Chapter 2: Problem 17

Find the slope-intercept form of the line which passes through the given points. \(P\left(\frac{1}{2}, \frac{3}{4}\right), Q\left(\frac{5}{2},-\frac{7}{4}\right)\)

Short Answer

Expert verified
The slope-intercept form is \( y = -\frac{5}{4}x + \frac{11}{8} \).

Step by step solution

01

Calculate the Slope

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting \(x_1 = \frac{1}{2}\), \(y_1 = \frac{3}{4}\), \(x_2 = \frac{5}{2}\), and \(y_2 = -\frac{7}{4}\), we get \[ m = \frac{-\frac{7}{4} - \frac{3}{4}}{\frac{5}{2} - \frac{1}{2}} = \frac{-\frac{10}{4}}{\frac{4}{2}} = \frac{-\frac{5}{2}}{2} = -\frac{5}{4}. \] So, the slope is \(-\frac{5}{4}\).
02

Write the Point-Slope Form Equation

Using the point-slope form of a line equation, \( y - y_1 = m(x - x_1) \), with point \(P\left(\frac{1}{2}, \frac{3}{4}\right)\) and slope \( m = -\frac{5}{4} \), the equation becomes \[ y - \frac{3}{4} = -\frac{5}{4}\left(x - \frac{1}{2}\right). \]
03

Simplify to Slope-Intercept Form

Expand the equation from step 2: \[ y - \frac{3}{4} = -\frac{5}{4}x + \frac{5}{8}. \] Adding \( \frac{3}{4} \) to both sides yields \[ y = -\frac{5}{4}x + \frac{5}{8} + \frac{3}{4}. \] Convert \( \frac{3}{4} \) to \( \frac{6}{8} \): \[ y = -\frac{5}{4}x + \frac{5}{8} + \frac{6}{8} = -\frac{5}{4}x + \frac{11}{8}. \] Therefore, the slope-intercept form is \( y = -\frac{5}{4}x + \frac{11}{8} \).

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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 the backbone of algebra and coordinate geometry. They describe the relationship between two variables in a straight line format. Often represented as \( y = mx + b \), this formula highlights two critical components:
  • \( m \), known as the slope, which indicates the steepness or inclination of the line.
  • \( b \), the y-intercept, which is the point where the line intersects the y-axis.
Linear equations are straightforward yet powerful, enabling us to predict values and understand trends in data. A solid grasp of linear equations is crucial for studies in many fields, including physics, economics, and social sciences.
Slope Calculation
Calculating the slope is an essential skill in determining the nature of a line connecting two points. The slope \( m \) is calculated by the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula finds the change in the y-values per unit change in the x-values.

This measure helps visualize how steep a line is. A positive slope means the line ascends as you move right, while a negative slope means it descends. If the slope is zero, the line is horizontal, and if undefined, the line is vertical. Knowing how to calculate the slope sets the foundation for understanding line behavior and trends in data sets.
Point-Slope Form
The point-slope form of a linear equation is another way to express the line's equation. It is particularly useful when you know a point on the line and the slope but not the full equation.

The formula is:\[ y - y_1 = m(x - x_1) \]Here,
  • \( (x_1, y_1) \) represents a specific point on the line.
  • \( m \) is the slope of the line.
To convert this form into the more familiar slope-intercept form, you can simplify and rearrange it into \( y = mx + b \). This method allows quick and efficient derivation of the equation of a line when starting from a known point and slope.
Coordinate Geometry
Coordinate geometry, or analytic geometry, involves studying geometry through algebra using the coordinate plane. It combines elements of algebra and geometry to solve geometric problems by using coordinates.
  • In a 2D plane, every point is represented as an ordered pair \( (x, y) \).
  • Lines are expressed through equations relating the x and y coordinates.
The use of coordinate geometry extends beyond just plotting points and lines. It is essential for understanding the spatial relationships and properties, like distance and midpoints. This approach creates a bridge between algebraic concepts and geometric intuition, allowing for a deeper comprehension of geometric principles through algebraic expressions.

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Most popular questions from this chapter

Solve the inequality. Write your answer using interval notation. \(x^{2} \geq|x|\)

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