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Chapter 4: Problem 4

$$ \text { For Problems } 4 \text { through } 13, \text { find the equation of the line with the given characteristics. } $$ $$ \text { Slope }-\frac{1}{2}, \text { passing through }(-2,-3) $$

Short Answer

Expert verified
The equation of the line with slope -1/2 passing through (-2,-3) is \(x + 2y + 8 = 0\).

Step by step solution

01

Substitute the given values

We substitute the given values into the point-slope formula. Since the slope \(m\) is \(-1/2\) and the point \((x_1, y_1)\) is \((-2, -3)\), after the substitution the equation is \(y - (-3) = -1/2 (x - (-2))\)
02

Simplify the equation

We simplify the equation to its canonical form. That involves removing the parentheses and simplifying like terms. First, \(y - (-3)\) simplifies to \(y + 3\) and \(x - (-2)\) simplifies to \(x + 2\). The equation becomes \(y + 3 = -1/2 (x + 2)\). Now to remove the fraction, we can multiply every term by 2 to obtain \(2y + 6 = -x - 2\). By rearranging the terms, we get \(x + 2y + 8 = 0\)
03

Final format

Typically, linear equations are written in the form \(Ax + By + C = 0\), where \(A,B,C\) are integers and \(A > 0\). In our case, we have \(x + 2y + 8 = 0\), which corresponds to the required format.

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Key Concepts

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

Understanding Linear Equations
Linear equations are foundational for understanding algebra and many areas of mathematics. They represent a relationship between two variables, usually x and y, where the graph of this relationship is a straight line. Visually, this means that for any two points on the line, the slope or the steepness of the line is constant. A linear equation can be written in several forms, including the standard form \(Ax + By = C\), where A, B and C are real numbers and x and y are variables.

When graphing a line with a given slope and a point through which it passes, as in the provided exercise, the point-slope form \(y - y_1 = m(x - x_1)\) is particularly useful. Here, \(m\) is the slope and \((x_1, y_1)\) is the point on the line. This form can instantly tell you how to draw the line if you know just one point on it and its slope.
Simplifying Equations
Simplifying equations is a crucial skill in algebra that makes them easier to interpret or solve. While simplifying, you combine like terms, remove parentheses, and solve fractions to achieve a simpler form. The student practice of taking a complicated expression like \(y - (-3) = -\frac{1}{2}(x - (-2))\) from the problem and transforming it to \(x + 2y + 8 = 0\) requires these steps. This process often involves distributing multiplication over addition or subtraction, as seen when the \(-\frac{1}{2}\) distributes across \((x + 2)\), and then moving terms to one side to isolate variables. Simplifying reveals the structure of the equation and prepares it for graphing or other types of analysis.
Canonical Form of a Line
The canonical form of a line, which is also known as standard form, can be written as \(Ax + By + C = 0\), where A, B, and C are integers, and A should be a positive integer if possible. This form is preferred because it avoids fractions and clearly delineates the linear relationship. From the step-by-step solution, our final equation \(x + 2y + 8 = 0\) fits this canonical form structure. It succinctly displays all the information needed to graph the line and is often used for theoretical purposes or in systems of equations because of its straightforwardness.

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Most popular questions from this chapter

It is \(10: 30\) A.M. Over the past half hour six customers have walked into the corner delicatessen. How many people might the owner expect to miss if he were to close the deli to run an errand for the next 15 minutes? Upon what assumption is this based? Suppose that between \(9: 30\) A.M. and \(11: 30\) A.M. he had 24 customers. Is it reasonable to assume that between 11:30 A.M. and 1:30 P.M. he will have 24 more customers? Why or why not?

At 8:00 A.M., a long-distance runner has run 10 miles and is tiring. She runs until \(9: 00\) A.M. but runs more and more slowly throughout the hour. By \(9: 00\) she has run 16 miles. (a) Sketch a possible graph of distance traveled versus time on the interval from \(8: 00\) to \(9: 00\). What are the key characteristics of this graph? (b) Suppose that at 8:00 A.M. she is running at a speed of 9 miles per hour. Find good upper and lower bounds for the total distance she has run by \(8: 30\) A.M. Explain your reasoning with both words and a graph.

Stories are told about some of the less fair-minded teams of early baseball (e.g., the Baltimore Orioles of the 1890 s) freezing baseballs until shortly before game time so that although the cover would feel normal, the core of the ball would be much colder. Then, they would attempt to introduce these balls into play when the opposing team was at bat, working on the assumption that the frozen balls would not travel as far when hit. Experiments have shown that a ball whose temperature is \(-10^{\circ} \mathrm{F}\) would travel 350 feet after a given swing of the bat, while a ball whose temperature is \(150^{\circ} \mathrm{F}\) would be hit 400 feet by the same swing. Assume this relationship is linear. Let \(B(T)\) be the distance this swing would produce, where \(T\) is the temperature in degrees Fahrenheit. (a) Find an equation for \(B(T)\). (b) What is the \(B\) -intercept? What is its practical meaning? (c) What is the slope of \(B(T) ?\) What is its practical meaning?

For Problems 14 through 16, find the slope of the secant line passing through points \(P\) and \(Q\), where \(P\) and \(Q\) are points of the graph of \(f(x)\) with the indicated \(x\) -coordinates. \(f(x)=\frac{3}{x}+2 x ;\) the \(x\) -coordinates of \(P\) and \(Q\) are \(x=3\) and \(x=3+k(k \neq 0)\), respectively.

find the slope of the secant line passing through points \(P\) and \(Q\), where \(P\) and \(Q\) are points of the graph of \(f(x)\) with the indicated \(x\) -coordinates. \(f(x)=a x^{2}+b x+c ;\) the \(x\) -coordinates of \(P\) and \(Q\) are \(x=k\) and \(x=k+h(h \neq 0)\), respectively.
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