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Chapter 3: Problem 58

Find an equation of the line passing through the given points. (a) Write the equation in standard form. (b) Write the equation in slope-intercept form if possible. $$ (-2,5) \text { and }(-8,1) $$

Short Answer

Expert verified
Slope-intercept form: \[ y = \frac{2}{3}x + \frac{19}{3} \], Standard form: \[ -2x + 3y = 19 \].

Step by step solution

01

- Find the Slope

Calculate the slope (\text{m}) of the line using the formula \text{m} = \frac{y_2 - y_1}{x_2 - x_1}. For the points (-2,5) and (-8,1): \[\text{m} = \frac{1 - 5}{-8 - (-2)} = \frac{-4}{-6} = \frac{2}{3}.\]
02

- Use Point-Slope Form

Use the point-slope form of the equation, which is y - y_1 = m(x - x_1). Using the slope \text{m} = \frac{2}{3} and point (-2, 5): \[y - 5 = \frac{2}{3}(x + 2).\]
03

- Convert to Slope-Intercept Form

Convert the point-slope form to slope-intercept form (y = mx + b):\[y - 5 = \frac{2}{3}(x + 2) \rightarrow y - 5 = \frac{2}{3}x + \frac{4}{3} \rightarrow y = \frac{2}{3}x + \frac{4}{3} + 5 \rightarrow y = \frac{2}{3}x + \frac{4}{3} + \frac{15}{3} \rightarrow y = \frac{2}{3}x + \frac{19}{3}.\]
04

- Convert to Standard Form

Convert the slope-intercept form to standard form (Ax + By = C): \[ y = \frac{2}{3}x + \frac{19}{3} \rightarrow 3y = 2x + 19 \rightarrow -2x + 3y = 19.\]

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Key Concepts

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

slope
The slope is a measure of how steep a line is. It shows how much the y-value changes for a given change in the x-value. For our given points (-2, 5) and (-8, 1), we calculate the slope using the formula: \[ \text{Slope (m)} = \frac{y_2 - y_1}{x_2 - x_1} \] \ By plugging in the points: \[ m = \frac{1-5}{-8-(-2)} = \frac{-4}{-6} = \frac{2}{3} \] This tells us that for every 3 units the x-value moves horizontally, the y-value will move 2 units vertically. \ The slope is fundamental because it helps to write the equation of the line in various forms.
point-slope form
The point-slope form of a line's equation leverages the slope and any given point on the line. The formula is: \[ y - y_1 = m(x - x_1) \] Here, (x_1, y_1) is a specific point on the line and m is the slope. \ For our example, using the point (-2, 5) and m = \( \frac{2}{3} \): \[ y - 5 = \frac{2}{3}(x + 2) \] This equation shows how the y-values change relative to the x-values using the slope. Point-slope form is particularly useful when you have a point and slope but not the y-intercept.
slope-intercept form
Slope-intercept form is one of the most common ways to express a linear equation. The formula is: \[ y = mx + b \] Here, m is the slope and b is the y-intercept (where the line crosses the y-axis). \ From the point-slope form, we convert to: \[ y - 5 = \frac{2}{3}(x + 2) \] Simplifying this, we get: \[ y = \frac{2}{3}x + \frac{19}{3} \] The y-intercept (b) here is \( \frac{19}{3} \). \ Slope-intercept form makes it easy to understand the line's behavior and direction quickly.
standard form
Standard form compacts a linear equation into a specific arrangement: \[ Ax + By = C \] Where A, B, and C are integers, and A should be positive. \ From the slope-intercept form \( y = \frac{2}{3}x + \frac{19}{3} \), we multiply through by 3 to clear the fractions: \[ 3y = 2x + 19 \] Then, rearrange to: \[ -2x + 3y = 19 \] Converting between forms helps in various scenarios like solving systems of equations or understanding geometric properties.

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