Problem 9 Find the equation of the line de... [FREE SOLUTION]

91影视

Elementary Geometry for College Students

Daniel C. Alexander, Alexander Koeberlein

$Math Studyset 91影视 Explanations$ Math

5 Edition

Chapter 10: Problem 9

Find the equation of the line described. Leave the solution in the form $A x+B y=C$. The line has slope $m=-\frac{2}{3}$ and contains $(0,5)$.

Short Answer

Expert verified

The equation of the line is $2x + 3y = 15$.

Step by step solution

Identify the Given Information

We are provided with a slope, $m = -\frac{2}{3}$, and a point on the line, $(0,5)$. The point indicates that when $x = 0$, $y = 5$.

Use Point-Slope Form

The point-slope form of the equation of a line is $y - y_1 = m(x - x_1)$. Here, $(x_1, y_1)$ is the given point, which is $(0, 5)$, and $m$ is the slope. So, the equation becomes: \[y - 5 = -\frac{2}{3}(x - 0)\]

Simplify the Equation

Simplify the equation obtained from the point-slope form.\[y - 5 = -\frac{2}{3}x\] Add 5 to both sides to isolate $y$: \[y = -\frac{2}{3}x + 5\]

Convert to Standard Form $Ax + By = C$

To convert the equation to standard form, eliminate the fraction by multiplying the entire equation by 3:\[3y = -2x + 15\] Rearrange to get $Ax + By = C$: \[2x + 3y = 15\]

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Key Concepts

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

Slope

The slope of a line is a measure of its steepness and direction. It is denoted by the letter "m." In the context of a linear equation, the slope tells us how much the y-coordinate changes for a change in the x-coordinate. The formula for slope is given by:

$ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} $

For the line in our problem, the slope is $ m = -\frac{2}{3} $, which means that for every 3 units the line moves horizontally, it descends by 2 units vertically.
The sign of the slope indicates the direction of the line. A negative slope, like in this case, means the line is decreasing. The line will slant downwards as we move from left to right.

Point-Slope Form

The point-slope form of a linear equation is a powerful tool that helps us write the equation of a line when we're given the slope and one point on the line. The form is:

$ y - y_1 = m(x - x_1) $

Here, $(x_1, y_1)$ is a known point, and $m$ is the slope.
This formula is particularly useful when you need to find the equation of a line quickly without having to solve for the y-intercept explicitly. In the exercise, we utilize $(0,5)$ as our point $(x_1, y_1)$ and the slope $-\frac{2}{3}$ to form:

$ y - 5 = -\frac{2}{3}(x - 0) $

Simplifying leads us to a more straightforward linear expression, which is then easily converted into its other forms.

Standard Form

A linear equation in standard form is expressed as $ Ax + By = C $. Each of these represents integers, with $A$ typically being non-negative. This form is particularly useful for quickly identifying intercepts and establishing the relationships between variables.
To transform an equation from another form, like slope-intercept, into standard form:

Begin by clearing any fractions (often by multiplying through by the denominator).
Rearrange terms to align with $ Ax + By = C $.

In our problem, converting $ y = -\frac{2}{3}x + 5 $ involved multiplying the equation by 3 to clear the fraction and then rearranging to

$ 2x + 3y = 15 $

This conversion makes it easier to analyze and graph the equation.

Linear Equation

A linear equation forms a straight line when graphed on the coordinate plane. These equations are fundamental in algebra and have the general form of $ y = mx + b $, where $m$ is the slope and $b$ is the y-intercept.
Linear equations can be used in various forms:

Slope-Intercept Form: $ y = mx + b $
Point-Slope Form: $ y - y_1 = m(x - x_1) $
Standard Form: $ Ax + By = C $

Each form serves different purposes; slope-intercept is great for graphing, point-slope is excellent for quickly writing an equation from a point and slope, and standard form is ideal for systems of equations. In this exercise, we start with the point-slope form and derive the standard form, demonstrating how different conditions dictate which form is most useful.

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91影视

Find the equation of the line described. Leave the solution in the form \(A x+B y=C\). The line has slope \(m=-\frac{2}{3}\) and contains \((0,5)\).

Short Answer

Step by step solution

Identify the Given Information

Use Point-Slope Form

Simplify the Equation

Convert to Standard Form \(Ax + By = C\)

Key Concepts

Slope

Point-Slope Form

Standard Form

Linear Equation

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