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Chapter 9: Problem 24

In Exercises \(7-24,\) find the general form of the equation of the line satisfying the conditions given and graph the line. With \(x\) -intercept \((4,0)\) and \(y\) -intercept \((0,-3)\)

Short Answer

Expert verified
\(3x - 4y - 12 = 0\)

Step by step solution

01

Identify the intercepts

The problem gives two points where the line intersects the axes. These points are: - x-intercept: (4, 0) - y-intercept: (0, -3)
02

Determine the slope

Use the slope formula (m) which is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plug the points (4, 0) and (0, -3) into the formula: \[ m = \frac{-3 - 0}{0 - 4} = \frac{-3}{-4} = \frac{3}{4} \]So, the slope (m) is \( \frac{3}{4} \).
03

Write the slope-intercept form of the line

The slope-intercept form of a line is given by \[ y = mx + b \]where m is the slope and b is the y-intercept. We already have m as \( \frac{3}{4} \) and the y-intercept (0, -3) gives us b = -3. So, the equation becomes: \[ y = \frac{3}{4}x - 3 \]
04

Convert to general form

To convert the slope-intercept form into the general form Ax + By + C = 0, rearrange the terms: \[ y = \frac{3}{4}x - 3 \]Multiply every term by 4 to clear the fraction: \[ 4y = 3x - 12 \]Then rearrange to \[ 3x - 4y - 12 = 0 \]Thus, the general form is \[ 3x - 4y - 12 = 0 \]
05

Graph the line

Plot the intercepts on the graph: (4, 0) and (0, -3). Draw a line passing through these points. Ensure the line extends in both directions and intersects the axes at the given intercepts.

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Key Concepts

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

Understanding the x-intercept
The x-intercept of a line is where the line crosses the x-axis. At this point, the value of y is always 0. In our problem, the x-intercept is given as (4, 0). This means when x = 4, y = 0. Identifying the x-intercept is important because it helps us find a specific point on the line which we're looking for. Generally: If you have an equation of the line, to find the x-intercept, set y = 0 and solve for x.
Understanding the y-intercept
The y-intercept of a line is where the line crosses the y-axis. Here, the value of x is always 0. For this exercise, the y-intercept is given as (0, -3). In simpler terms, when x = 0, y = -3. Like the x-intercept, knowing the y-intercept helps us locate another definite point on the line. For any line equation, to find the y-intercept, set x = 0 and solve for y.
Calculating the slope using the slope formula
The slope of a line describes how steep the line is, and it can be determined using the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \) In our problem, we use the intercept points (4, 0) and (0, -3). Substituting these points into the formula gives us: \( m = \frac{-3 - 0}{0 - 4} = \frac{-3}{-4} = \frac{3}{4} \) So, the slope (m) is \( \frac{3}{4} \). Understanding slope helps us know the direction and steepness of the line.
Using the slope-intercept form
The slope-intercept form of a line is written as: \( y = mx + b \) Here, m is the slope, and b is the y-intercept. From the previous steps, we know that m is \( \frac{3}{4} \) and b = -3. So, substituting these values into the slope-intercept form, we get: \( y = \frac{3}{4}x - 3 \) This form is very helpful in quickly sketching a line and understanding its basic characteristics.
Transforming to the general form of a line
The general form of a linear equation is written as: \( Ax + By + C = 0 \) To convert from the slope-intercept form to the general form, rearrange the terms and clear any fractions. Starting from our equation in slope-intercept form: \( y = \frac{3}{4}x - 3 \) First, multiply every term by 4 to eliminate the fraction: \( 4y = 3x - 12 \) Next, move all terms to one side to set the equation to 0: \( 3x - 4y - 12 = 0 \) Thus, the equation in its general form is: \( 3x - 4y - 12 = 0 \) In this form, A = 3, B = -4, and C = -12.

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Most popular questions from this chapter

In Exercises \(1-6,\) find the slope of the line passing through the given pair of points. \((a, b)\) and \((c, d)\)

In Exercises \(43-46\), find the general form of the equation of the line perpendicular to the line that contains the given points and that passes through the point midway between them. \((2,-7)\) and \((6,1)\)

Find the general form of the equation of the line passing through \((4,-7)\) and perpendicular to the line with equation \(x-5 y+7=0\)

In Exercises \(7-24,\) find the general form of the equation of the line satisfying the conditions given and graph the line. With \(y\) -intercept \((0,4)\) and slope \(-3\)

In Exercises \(43-46\), find the general form of the equation of the line perpendicular to the line that contains the given points and that passes through the point midway between them. \((5,-3)\) and \((7,1)\)
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