Problem 39 Find the equation of the line th... [FREE SOLUTION]

Chapter 12: Problem 39

Find the equation of the line through the given points. $$(-5,1) \text { and }(2,-6)$$

Short Answer

Expert verified

The equation of the line passing through the points (-5,1) and (2,-6) is $ y = -x - 4 $

Step by step solution

Find the Slope

To find the slope (m), use the formula $\( m = \frac{y2 - y1}{x2 - x1} $\), where $(x1, y1)$ and $(x2, y2)$ are the coordinates of two points on the line. For the given points, $(-5, 1)$ and $(2, -6)$, this becomes: $ m = \frac{-6 - 1}{2 - (-5)} = -1 $.

Find the Y-Intercept

Use the formula $ b = y - mx $ to find the y-intercept. Substituting one of the given points and the slope into this formula, we get $ b = 1 - (-1)(-5) = -4 $.

Write the Equation of the Line

Now that we have the slope ($ m = -1 $) and y-intercept ($ b = -4 $), we can substitute these values into the slope-intercept equation to find: $ y = -1x - 4 $, or simplified as: $ y = -x - 4 $. This is the equation of the line passing through the given points.

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Key Concepts

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

Slope Calculation

The slope of a line is a measure of its steepness. You can think of it as a way to describe how much "rise" there is for a given "run". When you're given two points on a line, the slope can be calculated through a straightforward process.

To find the slope, use this formula:

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

Here, $(x_1, y_1)$ and $(x_2, y_2)$ represent two distinct points along the line.

In the original exercise, you have points $(-5, 1)$ and $(2, -6)$. Substituting these into the equation gives:

$ m = \frac{-6 - 1}{2 - (-5)} = \frac{-7}{7} = -1 $

This indicates a downward slope, where the line decreases by 1 unit vertically for each increase of 1 unit horizontally.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. It tells you the value of $ y $ when $ x $ is zero (0). This is a key component of the line's equation, giving you an anchor point.

To find the y-intercept $ b $, you can use the formula:

$ b = y - mx $

Pick one of the points, say $(-5, 1)$, and use the slope you calculated.

So, you'll have:

$ b = 1 - (-1)(-5) = 1 - 5 = -4 $

This result tells us that the line intercepts the y-axis at $ -4 $, providing an important point for graphing or visualizing the line.

Slope-Intercept Form

The slope-intercept form is a way to write the equation of a line using its slope and y-intercept. It is very popular because it's straightforward and descriptive.

This form is represented as:

$ y = mx + b $

- Where $ m $ is the slope and $ b $ is the y-intercept.

In the original example, your slope $ (m) $ is $-1$ and the y-intercept $ (b) $ is $-4$. Plug these values into the slope-intercept equation:

$ y = -1x - 4 $ or simplified to $ y = -x - 4 $

This equation tells you how the line behaves, making it easy to predict $ y $ for any $ x $, graph the line, or understand its characteristics.

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91影视

Find the equation of the line through the given points. $$(-5,1) \text { and }(2,-6)$$

Short Answer

Step by step solution

Find the Slope

Find the Y-Intercept

Write the Equation of the Line

Key Concepts

Slope Calculation

Y-Intercept

Slope-Intercept Form

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