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Chapter 3: Problem 68

Find the slope of the line that passes through the given points. $$(-4,1),(-1,4)$$

Short Answer

Expert verified
The slope is 1.

Step by step solution

01

Identify the given points

The given points are (-4,1) and (-1,4). Assign them as (x1, y1) and (x2, y2) respectively.
02

Substitute the values into the slope formula

Substitute the values into the slope formula \(\frac{y_2 - y_1}{x_2 - x_1}\) which gives us \(\frac{4 - 1}{-1 - (-4)}\).
03

Calculate the expression

Calculate the expression which gives \(\frac{3}{3}\).

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Key Concepts

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

Coordinate Geometry
Coordinate geometry is the method of representing geometric shapes in a numeric way. This allows us to analyze geometric relationships using algebra. By assigning a coordinate grid to a plane, each point on that plane can be given a unique pair of numbers, known as coordinates. Typically, these coordinates are in the form of \(x, y\).

  • The first number, \((x)\), is called the 'x-coordinate,' and it tells us how far right (or left) the point is from the vertical axis (y-axis).
  • The second number, \((y)\), is called the 'y-coordinate,' signifying how high (or low) the point is from the horizontal axis (x-axis).
To find the slope of the line that passes through two points, we use these coordinates. This approach makes use of both algebraic and geometric principles to analyze and solve problems involving lines, shapes, and angles.
Slope Formula
The slope formula is used to calculate the steepness or incline of a line. Knowing how to apply this formula is crucial for understanding how much a line rises or falls between two points. The slope \(m\) of a line that passes through two points \( (x_1, y_1) \) and \((x_2, y_2)\) is given by:

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

What this formula does is measure the vertical change (\

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Most popular questions from this chapter

Graph each equation using any method. $$x=0$$

Graph each equation using any method. $$3 y=7$$

From geometry, we know that two points determine a line. Explain why it is good practice when graphing linear equations to find and plot three points instead of just two.

Set up a variation equation and solve for the requested value. The interest earned on a fixed amount of money varies jointly with the annual interest rate and the time that the money is left on deposit. If an account earns \(120\)dollar at \(8 \%\) annual interest when left on deposit for 2 years, how much interest would be earned in 3 years at an annual rate of \(12 \% ?\)

Express each joint variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 3. (OBJECTIVE 3) \(y\) varies jointly with \(r\) and \(s .\) If \(y=4\) when \(r=2\) and \(s=6,\) find \(y\) when \(r=3\) and \(s=4\).
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