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Chapter 3: Problem 5

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Slope \(=-8,\) passing through \((-3,-2)\)

Short Answer

Expert verified
The slope-intercept form of the equation for the given conditions is \(y = -8x - 26\).

Step by step solution

01

Apply the point and slope to the formula

Substitute the given values into the point-slope form equation. The slope \(m = -8\) and the point is \((-3, -2)\). So, \(y - (-2) = -8(x -(-3))\). Simplifying this gives us \(y + 2 = -8(x + 3)\).
02

Distribute multiplication

Now, distribute the multiplication to obtain \(y + 2 = -8x - 24\).
03

Transform to slope-intercept form

Lastly, convert to the slope-intercept form of line equation by isolating \(y\). By subtracting 2 from both sides, we obtain \(y = -8x -26\).

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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 linear equation is one of the most popular ways to express the equation of a line. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept, which is the point where the line crosses the y-axis. This form is incredibly handy because it allows you to quickly identify both the slope and the y-intercept of the line just by looking at the equation.How to Identify Key Features:
  • Slope \( (m) \): This tells you how steep the line is. A positive slope means the line rises as it moves to the right, while a negative slope means it falls.
  • Y-Intercept \( (b) \): This is the value of \( y \) when \( x = 0 \). It tells you where the line crosses the y-axis.
This form is not only helpful in graphing but also when writing equations based on given data points, like moving from point-slope form to slope-intercept form.
Linear Equations
Linear equations are algebraic expressions of lines in a two-dimensional space and they show a direct relationship between two variables, traditionally \( x \) and \( y \). These relationships are expressed with equations of the highest degree of one, such as \( y = mx + b \). They always form straight lines when graphed on a coordinate plane.Key Characteristics:
  • Simplicity: They can often be solved for one variable in terms of the other and are easy to graph.
  • Forms: While the slope-intercept form \( y = mx + b \) is common, other forms include standard form \( Ax + By = C \), and point-slope form \( y - y_1 = m(x - x_1) \).
  • Applications: These equations are used for real-world phenomena like predictive modeling, economics, and science.
Each linear equation can be transformed between forms for various uses and simplifications, making them versatile tools in algebra.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, involves using algebra to study geometric problems. It allows us to examine figures in a coordinate plane using a pair of numerical coordinates, which provide a bridge between algebraic equations and geometric shapes.Basic Elements:
  • Coordinates: Each point on the plane is represented by an ordered pair \((x, y)\).
  • Lines: Defined by linear equations, lines can be analyzed for specific properties such as requiring determination of slopes, midpoints, and lengths.
  • Transformation: Shifting between different forms of linear equations to suit various problems is a key skill.
A solid understanding of coordinate geometry provides the groundwork for higher mathematics, such as calculus and helps in solving real-world problems by using of graphical representation.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The inequality \(2 x-3 y<6\) contains a "less than" symbol, so its graph lies below the boundary line.

Solve for \(y\) and put the equation in slope-intercept form. $$y+3=-\frac{3}{2}(x-4)$$

A new car worth \(\$ 24,000\) is depreciating in value by \(\$ 3000\) per year. The mathematical model $$y=-3000 x+24,000$$ describes the car's value, \(y,\) in dollars, after \(x\) years. a. Find the \(x\)-intercept. Describe what this means in terms of the car's value. b. Find the \(y\)-intercept. Describe what this means in terms of the car's value. c. Use the intercepts to graph the linear equation. Because \(x\) and \(y\) must be nonnegative (why?), limit your graph to quadrant I and its boundaries. d. Use your graph to estimate the car's value after five years.

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-\frac{5}{2} x+1$$

The graph shows that in \(2000,45 \%\) of U.S. adults believed that most qualified students get to attend college. For the period from 2000 through 2010 , the percentage who believed that a college education is available to most qualified students decreased by approximately 1.7 each year. These \- conditions can be described by the mathematical model $$ Q=-1.7 n+45 $$ where \(Q\) is the percentage believing that a college \- education is available to most qualificd students \(n\) years after 2000 a. Let \(n=0,5,10,15,\) and \(20 .\) Make a table of values showing five solutions of the equation. b. Graph the formula in a rectangular coordinate system. Suggestion: Let each tick mark on the horizontal axis, labeled \(n\), represent 5 units. Extend the horizontal axis to include \(n=25 .\) Let each tick mark on the vertical axis, labeled \(Q\), represent 5 units and extend the axis to include \(Q=50\) \- c. Use your graph from part (b) to estimate the percentage of U.S. adults who will believe that a college education is available to most qualified students in 2018 . \- d. Use the formula to project the percentage of U.S. adults who will believe that a college education is available to most qualified students in 2018 .
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