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Chapter 3: Problem 67

Find an equation of the line that satisfies the given conditions. (a) Write the equation in slope-intercept form. (b) Write the equation in standard form. Through \((7,2) ;\) parallel to \(3 x-y=8\)

Short Answer

Expert verified
Slope-intercept form: y = 3x - 19. Standard form: 3x - y = 19.

Step by step solution

01

Identify the slope of the given line

The given line equation is in the form of standard form, which is: 3x - y = 8. To find the slope, convert it to slope-intercept form: y = mx + b. Let's solve for y:-y = -3x + 8.Multiply by -1:y = 3x - 8. So, the slope (m) is 3.
02

Use point-slope form for the new line

The point-slope form of a line is given by: y - y_1 = m(x - x_1).We have the point (7, 2) and the slope m = 3. Substitute these values into the formula: y - 2 = 3(x - 7).
03

Solve for y to convert to slope-intercept form

Expand and simplify the equation: y - 2 = 3x - 21.Add 2 to both sides: y = 3x - 19.So, the slope-intercept form is: y = 3x - 19.
04

Convert the slope-intercept form to standard form

Starting from the slope-intercept form: y = 3x - 19, let's convert it into the standard form (Ax + By = C):Move 3x to the left side: -3x + y = -19.Multiplying the entire equation by -1 to make the x term positive:3x - y = 19.So, the standard form is: 3x - y = 19.

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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 line equation is a convenient way to describe a line on a graph. It is written as:

\( y = mx + b \)

Key elements here are:
  • \( m \) represents the slope of the line, which shows how steep the line is.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
To find the slope-intercept form from a standard equation like \( 3x - y = 8 \), you need to solve for \( y \). This was done by isolating \( y \) on one side:

\( y = 3x - 8 \).

So, the slope \( m \) is 3, and the y-intercept \( b \) is -8.
standard form
The standard form of a line equation is another common way to write the equation of a line. It is represented as:

\( Ax + By = C \)

Here:
  • \( A \) and \( B \) are coefficients of \( x \) and \( y \), respectively.
  • \( C \) is a constant.
  • \( A \) should not be negative, although different conventions may allow for it.
Converting from slope-intercept form to standard form involves rearranging the equation. Starting from \( y = 3x - 19 \), we move the \( 3x \) term by subtracting it from both sides to get:

\( -3x + y = -19 \).

Finally, we'll multiply the whole equation by \( -1 \) to make the \( x \) term positive, resulting in:

\( 3x - y = 19 \).
point-slope form
The point-slope form is especially useful for writing the equation of a line when you know a point on the line and the slope. It is expressed as:

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

Here:
  • \( m \) is the slope of the line.
  • (\( x_1 \), \( y_1 \)) is a specific point on the line.
In our exercise, the point given is (7, 2) and the slope \( m \) is 3. Plugging these values into the formula, we get:

\( y - 2 = 3(x - 7) \).

This form can then be rearranged to slope-intercept form by isolating \( y \).
parallel lines
Parallel lines have identical slopes but different y-intercepts. This means that while they never meet, they extend in the same direction.

In mathematical terms, if two lines are parallel, their slopes \( m_1 \) and \( m_2 \) are equal:

\( m_1 = m_2 \).

For the given problem, the initial line's equation is \( 3x - y = 8 \). Converting this to slope-intercept form, we found the slope as 3.

Any line parallel to this one must also have a slope of 3.

Hence, using the point (7, 2) and the slope 3 in point-slope form helps us write an equation for the new, parallel line.

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Most popular questions from this chapter

Write an equation in the form \(y=m x\) for each situation. Then give the three ordered pairs associated with the equation for x-values 0,5, and 10. See Example 7(a). \(x\) represents the number of credit hours taken at Kirkwood Community College at \(\$ 111\) per credit hour, and \(y\) represents the total tuition paid for the credit hours (in dollars).

If a line has slope 0.2, then any line parallel to it has slope _____, and any line perpendicular to it has slope _____.

Write each inequality or compound inequality using interval notation. See Sections 1.1 and 2.6. $$ -4 \leq x \leq 4 $$

Three points that lie on the same straight line are said to be collinear. Consider the points \(A(3,1), B(6,2),\) and \(C(9,3) .\) Find the slope of segment \(A C\).

Find the midpoint of each segment with the given endpoints. $$ (5,2) \text { and }(-1,8) $$
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