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Chapter 1: Problem 53

Find an equation of the line that passes through the given point and has the indicated slope \(m\). Sketch the line.$$\left(4, \frac{5}{2}\right), \quad m=0$$

Short Answer

Expert verified
The equation of the line that passes through the point \((4, \frac{5}{2})\) and has a slope of 0 is \(y = \frac{5}{2}\).

Step by step solution

01

Identify the Given Components

The given point is \((4, \frac{5}{2})\) and the slope of the line \(m\) is given as 0.
02

Insert the Given Values Into the Equation

You use the slope-intercept form of the line which is \(y = mx + c\). Since the slope \(m\) is 0, the equation simplifies to \(y = c\). Here, \(c\) is the y-intercept. The line passes through the point \((4, \frac{5}{2})\), so the y-intercept is \(\frac{5}{2}\). Therefore, the required equation of the line is \(y = \frac{5}{2}\).
03

Sketching the Line

On a graph, locate the y-coordinate at \(\frac{5}{2}\). Since the slope is 0, this will result in a straight horizontal line.

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Key Concepts

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

Understanding the Slope-Intercept Form
When we talk about linear equations, one of the most common forms to express them is the **slope-intercept form**. This form allows us to quickly identify two critical components of a line: the slope and the y-intercept. The general formula for this form is:
\[y = mx + c\]where:
  • **\(y\)** is the dependent variable, affected by changes in \(x\) and line position.
  • **\(m\)** represents the slope of the line. This tells us how steep the line is.
  • **\(x\)** is the independent variable, whose value can be freely chosen.
  • **\(c\)** is the y-intercept of the line, indicating where the line crosses the y-axis.
The slope-intercept form is helpful because:
  • It clearly shows how changes in \(x\) might affect \(y\), thanks to the slope \(m\).
  • We can see where the line intersects the y-axis due to the y-intercept \(c\).
  • It's a direct way to express most linear equations and easy to graph.
Characteristics of a Horizontal Line
A **horizontal line** in a coordinate system might seem simple, but understanding its properties is essential. When a line is horizontal:
  • The slope \(m\) is always zero. This is because there is no vertical change as you move along the line, regardless of how far you travel.
  • The equation of a horizontal line can be expressed in the form \(y = c\), meaning \(y\) remains constant.
  • This constancy means that the line is parallel to the x-axis.
For instance, when a point like \((4, \frac{5}{2})\) lies on a line with a slope of zero, the resulting line has the equation:
\[y = \frac{5}{2}\]This maintains a consistent y-value of \(\frac{5}{2}\) across all x values.
Horizontal lines are straightforward to graph:
  • Locate the point where \(y = \frac{5}{2}\) on the y-axis.
  • Draw a line parallel to the x-axis passing through this point.
Significance of the Y-Intercept
The **y-intercept** is a key point in any linear equation as it provides a starting reference on the graph. This point occurs where the line crosses the y-axis. In the slope-intercept equation \(y = mx + c\), the **y-intercept** is represented by \(c\). It signifies:
  • The value of \(y\) when \(x\) is zero. This gives you a fixed starting point for graphing the line.
  • A reference point that can help verify the line's accuracy when constructing the graph.
For horizontal lines like the example before, the y-intercept is essentially the only feature defining the line, since every point on the line shares this \(y\)-value.
Understanding the y-intercept is vital for:
  • Quickly graphing linear equations.
  • Interpreting the behavior of the line in practical applications (e.g., initial values in a financial situation).

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer.If the inverse function of \(f\) exists and the graph of \(f\) has a \(y\) -intercept, then the \(y\) -intercept of \(f\) is an \(x\) -intercept of \(f^{-1}\).

Fill in the blanks. The mathematical model \(y=\frac{k}{x}\) is an example of _____ variation.

The lengths (in feet) of the winning men's discus throws in the Olympics from 1920 through 2008 are given by the following ordered pairs. $$\begin{aligned} &(1920,146.6) \quad(1956,184.9) \quad(1984,218.5)\\\ &(1924,151.3) \quad(1960,194.2) \quad(1988,225.8)\\\ &(1928,155.3) \quad(1964,200.1) \quad(1992,213.7)\\\ &(1932,162.3) \quad(1968,212.5) \quad(1996,227.7)\\\ &(1936,165.6) \quad(1972,211.3) \quad(2000,227.3)\\\ &\begin{array}{lll} (1948,173.2) & (1976,221.5) & (2004,229.3) \\ (1952,180.5) & (1980,218.7) & (2008,225.8) \end{array} \end{aligned}$$ (a) Sketch a scatter plot of the data. Let \(y\) represent the length of the winning discus throw (in feet) and let \(t=20\) represent 1920 (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c).

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(P\) varies directly as \(x\) and inversely as the square of \(y\) \(\left(P=\frac{28}{3} \text { when } x=42 \text { and } y=9 .\right)\)

Use Hooke's Law for springs, which states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. The coiled spring of a toy supports the weight of a child. The spring is compressed a distance of 1.9 inches by the weight of a 25 -pound child. The toy will not work properly if its spring is compressed more than 3 inches. What is the maximum weight for which the toy will work properly?
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