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Chapter 1: Problem 24

Find an equation of the line: with slope \(-5\), containing \((5,0)\).

Short Answer

Expert verified
The equation of the line is y = -5x + 25.

Step by step solution

01

Identify the given values

The given slope is -5and the line contains the point (5,0).
02

Use the point-slope form equation

The point-slope form of a line's equation is y - y_1 = m(x - x_1) . Substitute m = -5y_1 = 0x_1 = 5.
03

Substitute values and simplify

Substitute my_1 x_1, into the equation: y - 0 = -5(x - 5)Linear Equation is: y = -5x + 25.

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Key Concepts

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

Point-Slope Form
When you need to find the equation of a line, the point-slope form is a handy tool. It's written as \(y - y_1 = m(x - x_1)\). Here, \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. This form is very useful because you can directly plug in the values you know to get the equation of your line. For instance, in the exercise, we have \(m = -5 \), \(x_1 = 5 \), and \(y_1 = 0 \). This translates to \(y - 0 = -5(x - 5)\).
Slope
The slope of a line tells you how steep the line is. It's a measure of the vertical change for every unit of horizontal change. Mathematically, it is represented as \( m = \frac{rise}{run} \). For example, a slope of \(-5\) means that for every 1 unit you move to the right, the line goes down by 5 units. In the exercise, we used this value directly in the point-slope formula.
Substitution Method
The substitution method involves replacing variables with known values. This is often used to find the specific equation of a line once you have the general form. In the exercise, we substituted \(m\), \(x_1\), and \(y_1\) into the point-slope equation. Specifically, we substituted \m = -5\, \x_1 = 5\, and \y_1 = 0\ into \(y - y_1 = m(x - x_1)\). This gave us \(y - 0 = -5(x - 5)\).
Simplifying Equations
Simplifying equations makes them easier to understand and use. Once you've substituted your values into the point-slope form, you should simplify it to get the line's equation in slope-intercept form \(y = mx + b\). In the exercise, we started with \(y - 0 = -5(x - 5)\). To simplify, we just performed the multiplication and addition: \(y = -5x + 25\).

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