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

Chapter 1: Problem 9

Find the equation of the line that passes through the two given points. Write the line in slope-intercept form $(y=m x+c)$, if possible. $$(-9,1) \text { and }(3,-7)$$

Short Answer

Expert verified

The equation is $y = -\frac{2}{3}x - 5$.

Step by step solution

Determine the Slope

To find the equation of a line that passes through two points \, (-9, 1) and (3, -7) , we must first calculate the slope $m$ of the line. The slope $m$ is given by the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$. Substitute the given points into the formula: $m = \frac{-7 - 1}{3 - (-9)} = \frac{-8}{3 + 9} = \frac{-8}{12}$. Simplifying this gives us the slope: $m = -\frac{2}{3}$.

Use the Point-Slope Form

With the slope $m = -\frac{2}{3}$, use one of the given points to write the equation of the line in point-slope form: $y - y_1 = m(x - x_1)$. Let's use point (-9,1): Substitute $m = -\frac{2}{3}$, $x_1 = -9$, and $y_1 = 1$, we get $y - 1 = -\frac{2}{3}(x + 9)$.

Simplify to Slope-Intercept Form

Now, convert the point-slope form into the slope-intercept form $y = mx + c$. From $y - 1 = -\frac{2}{3}(x + 9)$, distribute $-\frac{2}{3}$: $y - 1 = -\frac{2}{3}x - 6$. Add 1 to both sides to solve for $y$: $y = -\frac{2}{3}x - 6 + 1$. This simplifies to $y = -\frac{2}{3}x - 5$.

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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 fundamental concept in algebra that expresses the relationship between the coordinates on a Cartesian plane. It shows how the value of one variable depends on the value of another. The most popular form of the equation of a line is the slope-intercept form:

Slope-Intercept Form: This form is written as $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept, which is the point where the line crosses the y-axis.

Other forms include the standard form and point-slope form, which have their unique uses. Understanding how to derive and manipulate these equations, especially how to transition between forms, is key to mastering algebra.

Slope Calculation

Calculating the slope $m$ is the first step in finding the equation of a line that passes through two given points. The slope measures the steepness and the direction of a line. Mathematically, it鈥檚 the "rise over run," or the vertical change over horizontal change. The formula for slope is:

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

Where $ (x_1, y_1) $ and $ (x_2, y_2) $ are the coordinates of the points. By substituting the values from the points $(-9, 1)$ and $(3, -7)$, you can calculate the slope:

$ m = \frac{-7 - 1}{3 - (-9)} = \frac{-8}{12} = -\frac{2}{3}$.

The slope $-\frac{2}{3}$ indicates the line decreases as it moves from left to right.

Point-Slope Form

The point-slope form is a convenient way to write the equation of a line when you know the slope and one point on the line. It is written as:

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

where $ (x_1, y_1) $ is a point on the line and $ m $ is the slope. This format emphasizes how a specific line relates to a particular point. You can use this form to quickly draft the general structure of the line. For instance, substituting our previously found slope, $-\frac{2}{3}$, and using the point $(-9, 1)$, the point-slope equation becomes:

$ y - 1 = -\frac{2}{3}(x + 9) $

This setting provides a straightforward method to then convert into other forms like the slope-intercept form.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a necessary skill in converting equations between forms. It involves performing operations such as distributing terms, combining like terms, and isolating variables.

Distributive Property: When you see an expression like $ a(b + c) $, you should multiply $a$ by each term inside the parentheses.
Combining Like Terms: Terms that have identical variables and powers can be combined. For instance, in the equation $y - 1 = -\frac{2}{3}x - 6$, simplifying to solve for $y$ requires combining constants.

Using these principles, simplify the point-slope equation $ y - 1 = -\frac{2}{3}(x + 9) $:

Distribute: $ y - 1 = -\frac{2}{3}x - 6 $
Add 1 to both sides: $ y = -\frac{2}{3}x - 5 $

This final expression $ y = -\frac{2}{3}x - 5 $ represents the line in slope-intercept form, showing the line's behavior clearly.

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91影视

Find the equation of the line that passes through the two given points. Write the line in slope-intercept form \((y=m x+c)\), if possible. $$(-9,1) \text { and }(3,-7)$$

Short Answer

Step by step solution

Determine the Slope

Use the Point-Slope Form

Simplify to Slope-Intercept Form

Key Concepts

Equation of a Line

Slope Calculation

Point-Slope Form

Simplifying Algebraic Expressions

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