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Chapter 3: Problem 36

Mixed Practice Find the slope of each line. See Examples 3 through 6. $$ 24 x-3 y=5.7 $$

Short Answer

Expert verified
The slope is 8.

Step by step solution

01

Identify the Standard Form

The given equation is \(24x - 3y = 5.7\). This equation is in standard form \(Ax + By = C\), with \(A = 24\), \(B = -3\), and \(C = 5.7\).
02

Convert to Slope-Intercept Form

To find the slope, we need to convert the equation to the slope-intercept form \(y = mx + b\). Start by isolating \(y\) on one side:\[-3y = -24x + 5.7\]
03

Solve for y

Divide each term by \(-3\) to solve for \(y\):\[y = \frac{-24}{-3}x + \frac{5.7}{-3}\]
04

Simplify the Equation

Simplify the fractions:\[y = 8x - 1.9\]This is now in the form \( y = mx + b \), where \( m \) is the slope.
05

Identify the Slope

In the equation \(y = 8x - 1.9\), the coefficient of \(x\) is the slope. Therefore, the slope \(m\) is 8.

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Key Concepts

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

Standard Form Equation
A standard form equation of a line is expressed as \(Ax + By = C\). This format is useful as it presents the coefficients of \(x\) and \(y\) clearly, which helps in analyzing the relationship between variables. In a standard form equation:
  • \(A\), \(B\), and \(C\) are integers.
  • \(A\) is usually a positive integer.
  • There are no fractions or decimals typically associated with \(A\), \(B\), and \(C\).
In our example, the equation \(24x - 3y = 5.7\) follows the standard format where \(A = 24\), \(B = -3\), and \(C = 5.7\). This structure, although straightforward, often requires conversion to other forms, like slope-intercept form, to easily evaluate properties like slope.
Slope-Intercept Form
To make equations more manageable and intuitive, they are often rewritten in slope-intercept form: \(y = mx + b\). This arrangement is particularly useful for determinining the slope and establishing the y-intercept. In slope-intercept form:
  • \(m\) is the slope of the line.
  • \(b\) is the y-intercept, indicating where the line crosses the y-axis.
For the equation \(y = 8x - 1.9\), we can immediately see:
  • The slope \(m\) is 8, meaning for every unit increase in \(x\), \(y\) increases by 8 units.
  • The y-intercept \(b\) is -1.9.
The slope-intercept form is instrumental for graphing linear equations quickly and understanding their behavior.
Isolating Variables
When converting equations into a more useful form, isolating variables is a critical step. This process involves manipulating an equation to get a particular variable by itself on one side. Here's how it works with the given equation:
  • Start with the standard form: \(24x - 3y = 5.7\).
  • Move terms involving \(x\) to the other side: \(-3y = -24x + 5.7\).
  • Divide each term by the coefficient of \(y\) (in this case, \(-3\)) to solve for \(y\).
Through isolating \(y\), the equation becomes clearer and allows for immediate identification of slope and intercepts. This approach efficiently transforms a cumbersome equation into a more interpretable format.
Simplifying Fractions
Simplifying fractions is an essential mathematical operation that makes expressions easier to work with. In our equation transformation:
  • The fraction \(\frac{-24}{-3}\) simplifies to 8, removing unnecessary complexity.
  • \(\frac{5.7}{-3}\) simplifies to -1.9, providing a clearer numerical value for addition or subtraction.

Both operations streamline algebraic expressions by reducing terms to their simplest form. This simplification is crucial for both computational ease and accurate interpretation when graphing or comparing lines. It forms the backbone of efficient algebraic manipulation, leading to more effective problem-solving.

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Most popular questions from this chapter

Write an equation in standard form of the line that contains the point (4,0) and is parallel to (has the same slope as) the line \(y=-2 x+3\)

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. The value of a building bought in 1995 may be depreciated (or decreased) as time passes for income tax purposes. Seven years after the building was bought, this value was \(\$ 225,000\) and 12 years after it was bought, this value was \(\$ 195,000\). a. If the relationship between number of years past 1995 and the depreciated value of the building is linear, write an equation describing this relationship. Use ordered pairs of the form (years past \(1995,\) value of building). b. Use this equation to estimate the depreciated value of the building in 2013 .

The amount \(y\) of land occupied by farms in the United States (in millions of acres) from 1997 through 2007 is given by \(y=-4 x+967\). In the equation, \(x\) represents the number of years after 1997 . (Source: National Agricultural Statistics Service) a. Complete the table. $$ \begin{array}{|c|c|c|c|} \hline \boldsymbol{x} & 4 & 7 & 10 \\ \hline \boldsymbol{y} & & & \\ \hline \end{array} $$ b. Find the year in which there were approximately 930 million acres of land occupied by farms. (Hint: Find \(x\) when \(y=930\) and round to the nearest whole number.) c. Use the given equation to predict when the land occupied by farms might be 900 million acres. (Use the hint for part b.)

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. In \(2003,\) there were 302 million magazine subscriptions in the United States. By 2007 , this number was 322 million. (Source: Audit Bureau of Circulation, Magazine Publishers Association) a. Write two ordered pairs of the form (years after \(2003,\) millions of magazine subscriptions) for this situation. b. Assume the relationship between years after 2003 and millions of magazine subscriptions is linear over this period. Use the ordered pairs from part (a) to write an equation for the line relating year after 2003 to millions of magazine subscriptions. C. Use this linear equation in part (b) to estimate the millions of magazine subscriptions in 2005 .

Solve. See the Concept Checks in this section. In general, what points can have coordinates reversed and still have the same location?
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