Problem 26 Find an equation of each line wi... [FREE SOLUTION]

Chapter 3: Problem 26

Find an equation of each line with the given slope that passes through the given point. Write the equation in the form $A x+B y=C .$ See Example 4. $$ m=-2 ; \quad(-11,-12) $$

Short Answer

Expert verified

The equation of the line is $2x + y = -34$.

Step by step solution

Understand the Point-Slope Form of a Line

To find the equation of a line given a slope $m$ and a point $(x_1, y_1)$, we use the point-slope form: \[ y - y_1 = m(x - x_1) \] For this exercise, the slope $m$ is $-2$, and the point is $(-11, -12)$. Thus, the point-slope equation becomes:\[ y + 12 = -2(x + 11) \]

Expand the Equation

Distribute the slope $-2$ through the expression on the right side of the equation:\[ y + 12 = -2x - 22 \]

Rearrange into Standard Form

The standard form of a line is $Ax + By = C$, where $A$, $B$, and $C$ are integers. To rearrange, first move all terms to one side of the equation:\[ 2x + y = -22 - 12 \]Combine like terms:\[ 2x + y = -34 \] This is the equation in the required form, $Ax + By = C$.

Verify the Solution

Substitute the point $(-11, -12)$ back into the equation to ensure it holds true:\[ 2(-11) + (-12) = -34 \]\[ -22 - 12 = -34 \]Since both sides are equal, the equation is verified.

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Key Concepts

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

Point-Slope Form

The point-slope form is a useful tool for writing the equation of a line when you know a point on the line and its slope. The formula is:

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

Here, $ m $ represents the slope of the line, and $ (x_1, y_1) $ is a known point on the line. This form is particularly handy because it directly incorporates both the slope and a specific point, simplifying the process of finding linear equations. For instance, if you have the slope $-2$ and the point $(-11, -12)$, substituting these into the point-slope form yields:

$ y + 12 = -2(x + 11) $

You now have an equation representing the line through that point with the specified slope.

Standard Form

The standard form of a linear equation is expressed as $ Ax + By = C $, where $ A $, $ B $, and $ C $ are integers, and $ A $ should not be negative. Converting an equation into standard form often involves rearranging and simplifying terms. Starting from the point-slope form, like $ y + 12 = -2x - 22 $, you need to distribute, collect all terms on one side, and arrange them:

Convert $ y + 12 = -2x - 22 $ to $ 2x + y = -34 $

This transformation maintains the relationship between $ x $ and $ y $ while presenting it in a format that is clear and organized, making calculations and graph interpretations easier. Remember that all coefficients should be integers.

Linear Equations

Linear equations represent relationships with constant rates of change and can be visually represented as straight lines on a graph. An equation for a line can take multiple forms, each conveying the same linear relationship in various ways. The general expression for a linear equation is:

$ Ax + By = C $

This equation tells us how $ y $ changes as $ x $ changes, characterized by an unchanging slope. Understanding linear equations is fundamental as they appear throughout algebra and in different real-world contexts, from calculating costs to determining velocity. Each form of a linear equation reveals insights into the line's properties, such as slope, intercepts, and angle with the axes.

Slope-Intercept Form

The slope-intercept form is widely used due to its straightforward representation, especially when graphing or identifying key line properties. This form is represented as:

$ y = mx + b $

Where $ m $ represents the slope, showing how steep the line is, and $ b $ denotes the y-intercept, the point where the line crosses the y-axis. To convert a linear equation to slope-intercept form, you simply solve for $ y $. For example, rearranging the equation $ 2x + y = -34 $ gives:

$ y = -2x - 34 $

This version makes graphing easy, as you can start at the y-intercept and use the slope to determine other points on the line, efficiently sketching the line's path on a graph.

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91影视

Find an equation of each line with the given slope that passes through the given point. Write the equation in the form \(A x+B y=C .\) See Example 4. $$ m=-2 ; \quad(-11,-12) $$

Short Answer

Step by step solution

Understand the Point-Slope Form of a Line

Expand the Equation

Rearrange into Standard Form

Verify the Solution

Key Concepts

Point-Slope Form

Standard Form

Linear Equations

Slope-Intercept Form

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