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Chapter 3: Problem 28

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example \(1 .\) Slope \(-\frac{5}{4},\) passes through \((2,0)\)

Short Answer

Expert verified
Equation in point-slope form: \( y = -\frac{5}{4}(x - 2) \), and in slope-intercept form: \( y = -\frac{5}{4}x + \frac{5}{2} \).

Step by step solution

01

Understand Point-Slope Form

The point-slope form of a line is written as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \((x_1, y_1)\) is a point on the line.
02

Substitute Given Values into Point-Slope Form

We have a slope \( m = -\frac{5}{4} \) and the point \((2, 0)\). Substitute these values into the point-slope form: \( y - 0 = -\frac{5}{4}(x - 2) \).
03

Simplify the Equation

Simplify the equation from Step 2: \( y = -\frac{5}{4}x + \frac{5}{4} \cdot 2 \). This yields \( y = -\frac{5}{4}x + \frac{5}{2} \).
04

Convert to Slope-Intercept Form

The slope-intercept form is \( y = mx + b \). From Step 3, the equation \( y = -\frac{5}{4}x + \frac{5}{2} \) is already in this form, with slope \( m = -\frac{5}{4} \) and y-intercept \( b = \frac{5}{2} \).

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Key Concepts

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

Equation of a Line
The equation of a line is a mathematical statement that expresses the relationship between the coordinates on a plane and describes a line on a two-dimensional graph. Understanding the equation of a line is crucial in geometry, especially when dealing with linear relationships.
  • There are different forms of the equation of a line, including the point-slope form, slope-intercept form, and standard form.
  • Each form is useful in particular situations and gives different insights into the properties of the line.
  • The equation helps in identifying important line characteristics such as slope and intercepts.
The point-slope form is particularly helpful when you know one point on the line and the slope. Slope-intercept form, on the other hand, is great for quickly identifying the slope and y-intercept without much computation.
Slope-Intercept Form
The slope-intercept form of a linear equation is incredibly user-friendly and frequently used because it straightforwardly reveals critical attributes of a line, specifically the slope and y-intercept. This form is expressed as \( y = mx + b \).
  • Here, \( m \) represents the slope of the line, which indicates how steep the line is. A positive slope means the line rises as you move from left to right, while a negative slope means it falls.
  • The \( b \) in the equation is the y-intercept. It tells you where the line crosses the y-axis.
The simplicity of this form makes it the go-to form for graphing lines or solving problems quickly. Here's a quick reminder: simply plug in any value for \( x \) to find the corresponding \( y \) using the slope-intercept form, which can then be easily plotted on a coordinate grid.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, connects algebra with geometry through graphs and figures described in coordinate planes. It is a tremendous asset in solving real-world problems by analyzing geometrical shapes and their properties numerically.
  • In this field, each point is described by an ordered pair \((x, y)\), which specifies its location on the Cartesian plane.
  • Lines are then defined based on these points and the relationships among them, often using equations like the ones discussed.
  • Coordinate geometry allows for a detailed analysis of lines, angles, and figures using a systematic and algebraic approach.
By using coordinate geometry, we can delve deeper into understanding both the equation of a line and its graphical representation. This helps in visualizing linear relationships and solving for unknowns using a blend of algebraic and graphical techniques.

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Most popular questions from this chapter

Find the slope of a line perpendicular to the line passing through the given two points. See Example \(9 .\) \(\left(-2, \frac{1}{2}\right)\) and \(\left(-1, \frac{3}{2}\right)\)

$$ \text { Simplify: } 40\left(\frac{3}{8} x-\frac{1}{4}\right)+40\left(\frac{4}{5}\right) $$

In your own words, what is a function?

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. See Example 8. \((11,0)\) and \((11,-5)\) \((14,6)\) and \((25,6)\)

Find the slope of each line. See Examples 4 and \(5 .\) $$2 y+2=-6$$
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