Problem 11 Find the equation of the line th... [FREE SOLUTION]

Chapter 1: Problem 11

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. $$(-12,-9) \text { and }(4,11)$$

Short Answer

Expert verified

The equation of the line is $y = \frac{5}{4}x + 6.$

Step by step solution

Calculate the Slope

To find the equation of a line in slope-intercept form, we first need to calculate the slope $m$ using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1, y_1) = (-12, -9)$ and $(x_2, y_2) = (4, 11)$. Substituting these values into the formula gives:\[m = \frac{11 - (-9)}{4 - (-12)} = \frac{11 + 9}{4 + 12} = \frac{20}{16} = \frac{5}{4}.\]So, the slope $m$ is $\frac{5}{4}.$

Use the Slope-Point Form

Using the slope-point form $ y - y_1 = m(x - x_1) $ with the slope $ m = \frac{5}{4} $ and one of the given points. Let's use $(x_1, y_1) = (4, 11)$:\[ y - 11 = \frac{5}{4}(x - 4). \]

Simplify to Slope-Intercept Form

Simplify the equation from Step 2 into the slope-intercept form $y = mx + c$:\[y - 11 = \frac{5}{4}x - \frac{5}{4} \times 4,\]\[y - 11 = \frac{5}{4}x - 5.\]Add 11 to both sides to isolate $y$:\[y = \frac{5}{4}x + 6.\]Thus, the equation of the line in slope-intercept form is $y = \frac{5}{4}x + 6.$

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Key Concepts

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

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most straightforward ways to express the equation of a line. It is written as:

$ y = mx + c $

In this format, $ m $ represents the slope of the line, while $ c $ represents the y-intercept. The y-intercept is the point where the line crosses the y-axis, and it tells us the value of $ y $ when $ x = 0 $. This form is particularly useful because it provides a quick visual understanding of a line's characteristic from its equation directly.
For example, with the equation $y = \frac{5}{4}x + 6$, we instantly know that:

The slope is $ \frac{5}{4} $
The y-intercept is 6, meaning the line crosses the y-axis at (0, 6).

Point-Slope Form

The point-slope form is a powerful way to write the equation of a line when it is easy for you to identify a slope and a specific point. It is expressed as:

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

Here, $(x_1, y_1)$ is a point on the line, and $m$ is the slope. The point-slope form is particularly helpful for constructing the equation of a line when you are given the slope and a point, or two points that you can use to find the slope.
For instance, if you have calculated the slope $m = \frac{5}{4}$ and have a known point like $(4, 11)$, you can plug these into the point-slope formula. This is why we derive:

$ y - 11 = \frac{5}{4}(x - 4) $

This form helps us transition smoothly into writing the equation in the slope-intercept form.

Calculating Slope

The slope of a line is crucial as it indicates the line's steepness and its direction (upwards or downwards). To calculate the slope when you have two points, use the formula:

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

Simply, $ y_1 $ and $ y_2 $ are the y-coordinates of two different points, and $ x_1 $ and $ x_2 $ are their respective x-coordinates. This formula calculates the "rise over run," or how much the line goes up or down for each unit it goes across.
For example, with points $(-12, -9)$ and $(4, 11)$, we determine:

$ m = \frac{11-(-9)}{4-(-12)} = \frac{20}{16} = \frac{5}{4} $

Such calculations help us not only write the line's equation but also understand the line's behavior relative to these markers.

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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. $$(-12,-9) \text { and }(4,11)$$

Short Answer

Step by step solution

Calculate the Slope

Use the Slope-Point Form

Simplify to Slope-Intercept Form

Key Concepts

Slope-Intercept Form

Point-Slope Form

Calculating Slope

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