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Chapter 5: Problem 11

Write an equation of the line satisfying the given conditions. Passing through \((-2,0)\) with slope \(-\frac{3}{4}\)

Short Answer

Expert verified
The equation of the line is \( y = -\frac{3}{4}x - \frac{3}{2} \).

Step by step solution

01

Understand the point-slope form

Recall the point-slope form of a line equation, which is given by: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.
02

Plug in the given point and slope

Plug in the point \((-2, 0)\) and slope \(-\frac{3}{4}\) into the point-slope form equation. Here, \( x_1 = -2 \), \( y_1 = 0 \), and \( m = -\frac{3}{4} \).\[ y - 0 = -\frac{3}{4}(x - (-2)) \]
03

Simplify the equation

Simplify the equation from Step 2:\[ y = -\frac{3}{4}(x + 2) \]Distribute the slope:\[ y = -\frac{3}{4}x - \frac{3}{4} \times 2 \]\[ y = -\frac{3}{4}x - \frac{3}{2} \]

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Key Concepts

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

Point-Slope Form
In linear equations, the point-slope form is essential. It helps in writing the equation of a line when you know a point and the slope. The formula is:
  • y - y_1 = m(x - x_1)
Here, \( m \) stands for the slope, and \( (x_1, y_1) \) represents a point on the line. For example, if a line passes through \( (-2, 0) \) and has a slope of \( -\frac{3}{4} \), you can plug these values into the formula.
  • When \( x_1 = -2 \) and \( y_1 = 0 \), the equation becomes: \( y - 0 = -\frac{3}{4}(x - (-2)) \).
  • This simplifies to \( y = -\frac{3}{4}(x + 2) \).
Point-slope form is particularly handy when you transform the equation into other formats, such as slope-intercept form.
Slope-Intercept Form
The slope-intercept form is another crucial representation of linear equations. This form makes it easy to identify the slope and y-intercept of a line. Its general formula is:ul>
  • y = mx + b
    • Here, \( m \) is the slope, and \( b \) is the y-intercept. Let's convert the point-slope form equation \( y = -\frac{3}{4}(x + 2) \) into the slope-intercept form.
    • First, distribute the slope: \( y = -\frac{3}{4}x - \frac{3}{2} \).
    • In this equation, \( -\frac{3}{4} \) is the slope, and \( -\frac{3}{2} \) is the y-intercept.
    This format is useful for quickly graphing the line and understanding its behavior.
    Coordinate Geometry
    Coordinate geometry connects algebra and geometry through graphs of equations. It's the study of geometric figures using the coordinate plane. Here, we find points based on their coordinates (x and y values). For instance, the point \( (-2, 0) \) in our exercise has its x-coordinate as -2 and y-coordinate as 0.
    • Using the point-slope form, we find how the line behaves around this point.
    • Slope helps determine how steep the line is and whether it goes up or down as you move along the x-axis.
    Coordinate geometry makes it easier to visualize equations and understand the spatial relationships between points, lines, and curves. It's essential for graphing and solving geometric problems algebraically.

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    Most popular questions from this chapter

    Sets of values are given for variables having a linear relationship. In each case, write the slope-intercept form for the equation of the line corresponding to the given set of values and answer the accompanying question. $$\begin{array}{|l|c|c|c|} \hline x \text { (Number of clerks working) } & 6 & 8 & 9 \\ \hline y \text { (Number of minutes waiting time) } & 8 & 5 & 3.5 \\ \hline \end{array}$$ What would the waiting time be if 4 clerks are working?

    Write an equation of the line satisfying the given conditions. Horizontal line passing through \((4,-3)\)

    Nature experts tell us that crickets can act as an outdoor thermometer because the rate at which a cricket chirps is linearly related to the temperature. At \(60^{\circ} \mathrm{F}\) crickets make an average of 80 chirps per minute, and at \(68^{\circ} \mathrm{F}\) they make an average of 112 chirps per minute. (a) Write an equation relating the average number of chirps \(c\) and the temperature \(t\) (b) What would be the average number of chirps at \(90^{\circ} \mathrm{F}\) ? (c) If you hear an average of 88 chirps per minute, what is the temperature? (d) Sketch a graph of this equation using the horizontal axis for \(t\) and the vertical axis for \(c\) (e) What is the \(t\) -intercept of this graph? What is the significance of this \(t\) -intercept?

    Round off to the nearest hundredth when necessary. Bridges (and many concrete highways) are constructed with "expansion joints," which are small gaps in the roadway between one section of the bridge and the next. These expansion joints allow room for the roadway to expand during hot weather. Suppose that a bridge has a gap of \(1.5 \mathrm{cm}\) when the air temperature is \(24^{\circ} \mathrm{C},\) that the gap narrows to \(0.7 \mathrm{cm}\) when the air temperature is \(33^{\circ} \mathrm{C},\) and that the width of the gap is linearly related to the temperature. (a) Write an equation relating the width of the gap \(w\) and the temperature \(t\) (b) What would be the width of a gap in this roadway at \(28^{\circ} \mathrm{C} ?\) (c) At what temperature would the gap close completely? (d) If the temperature exceeds the value found in part (c) that causes the gap to close, it is possible that the roadway could buckle. Is this likely to occur? Explain.

    Write an equation of the line satisfying the given conditions. Line has \(x\) -intercept \(-3\) and \(y\) -intercept 4
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