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Chapter 2: Problem 15

Use the point-slope formula to write an equation of the line having the given conditions. Write the answer in slope-intercept form (if possible). Passes through (3.4,2.6) and \(m=1.2\).

Short Answer

Expert verified
\( y = 1.2x - 1.48 \)

Step by step solution

01

Identify the Point-Slope Formula

The point-slope formula for a line is given by the equation: \[ y - y_1 = m(x - x_1) \] where - \(m\) is the slope of the line, - \( (x_1, y_1) \) is a point on the line.
02

Substitute the Given Values

We know the slope \( m = 1.2 \) and the point \( (x_1, y_1) = (3.4, 2.6) \). Substitute these values into the point-slope formula: \[ y - 2.6 = 1.2(x - 3.4) \]
03

Simplify the Equation

Distribute the slope on the right-hand side: \[ y - 2.6 = 1.2x - 4.08 \]
04

Solve for y to put in Slope-Intercept Form

Add 2.6 to both sides to isolate y: \[ y = 1.2x - 4.08 + 2.6 \] Combine like terms: \[ y = 1.2x - 1.48 \]

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Key Concepts

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

slope-intercept form
The slope-intercept form is a common way to represent linear equations. It is written as Default (GPT-3.5) Alternative (GPT-4) y = mx + b. Here,
  • m is the slope of the line: This number tells us how steep the line is
  • b is the y-intercept: This is where the line crosses the y-axis
To convert any linear equation to this form, you'll want to isolate y on one side of the equation. For example, in the solution of the exercise, we started with an equation in point-slope form: y - 2.6 = 1.2(x - 3.4) Distributing the 1.2 and then solving for y, we got y = 1.2x - 1.48, which is in the slope-intercept form. Knowing how to convert to slope-intercept form is very useful as it allows you to easily graph the line and quickly understand its properties.
linear equations
Linear equations represent straight lines and have the general form: ax + by = c In simpler terms, they depict relationships where there's a constant rate of change.

For our specific exercise, the linear equation was y = 1.2x - 1.48. Here:
  • 1.2 is the rate of change (or slope), meaning for every one unit increase in x, y increases by 1.2 units.
  • The constants may vary, but they always describe a straight line when plotted on a coordinate system.
It's essential to recognize that linear equations can sometimes show up in different forms, but they can always be rewritten or simplified to more familiar forms like slope-intercept form.
coordinate geometry
Coordinate geometry involves using algebraic equations to describe geometric shapes, like lines and circles. In our example, using the point (3.4, 2.6) and a slope of 1.2 involves placing these descriptions into a coordinate plane.

Here are some of the core takeaways:
  • Points on the Plane: Points are represented as (x, y) pairs, like (3.4, 2.6).
  • Plotting Lines: Lines can be plotted using their equations, and many lines can be described by their slopes and y-intercepts.
The way to represent a point through which a line passes and with a given slope as a linear equation is pivotal in understanding coordinate geometry deeply. Placing the line y - 2.6 = 1.2(x - 3.4) on a graph helps us visualize how lines and points connect.
algebraic manipulation
Algebraic manipulation involves rearranging and simplifying equations to make them easier to work with. In our exercise, we started with the point-slope form y - 2.6 = 1.2(x - 3.4) and rearranged it step-by-step to derive the slope-intercept form.

Here鈥檚 how it worked:
  • Substituted given values to get y - 2.6 = 1.2(x - 3.4).
  • Distributed the slope: y - 2.6 = 1.2x - 4.08.
  • Added 2.6: 1.2x - 4.08 + 2.6.
  • Combined like terms to isolate y: y = 1.2x - 1.48.
These steps demonstrate how algebraic manipulation helps in transforming equations into more useful forms, such as the slope-intercept form, making it easier to solve problems and understand their underlying meaning.

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