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Chapter 5: Problem 58

Write an equation of the line that passes through the point \((2,-1)\) and is parallel to the line whose equation is \(4 x-3 y=6\)

Short Answer

Expert verified
The equation of the line is \( y = \frac{4}{3}x - \frac{11}{3} \).

Step by step solution

01

- Identify the slope of the given line

The equation of the given line is \( 4x - 3y = 6 \). First, rewrite this equation in the slope-intercept form \( y = mx + b \) to find its slope. Solve for \( y \) by isolating it on one side of the equation:\[ 4x - 3y = 6 \]\[ -3y = -4x + 6 \]\[ y = \frac{4}{3}x - 2 \]So, the slope \( m \) of the given line is \( \frac{4}{3} \).
02

- Determine the slope of the parallel line

Since parallel lines have the same slope, the line passing through the point \((2, -1)\) will also have a slope of \( \frac{4}{3} \).
03

- Use the point-slope form

The point-slope form of the equation of a line is given by:\[ y - y_1 = m(x - x_1) \]Here, \( (x_1, y_1) = (2, -1) \) and \( m = \frac{4}{3} \). Substitute these values into the equation:\[ y - (-1) = \frac{4}{3}(x - 2) \]\[ y + 1 = \frac{4}{3}(x - 2) \]
04

- Simplify the equation

Distribute \( \frac{4}{3} \) on the right-hand side:\[ y + 1 = \frac{4}{3}x - \frac{8}{3} \]Subtract 1 from both sides:\[ y = \frac{4}{3}x - \frac{8}{3} - 1 \]\[ y = \frac{4}{3}x - \frac{8}{3} - \frac{3}{3} \]\[ y = \frac{4}{3}x - \frac{11}{3} \]So, the equation of the line is:\[ y = \frac{4}{3}x - \frac{11}{3} \]

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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 line's equation is very handy. It's expressed as \( y = mx + b \), where
  1. \( m \): the slope of the line, showing how steep it is.
  2. \( b \): the y-intercept, marking where the line crosses the y-axis.
For instance, in our exercise, we transformed \( 4x - 3y = 6 \) into slope-intercept form to find the slope. Here's how to do it:
  1. Isolate \( y \) on one side of the equation: \( -3y = -4x + 6 \).
  2. Divide by \( -3 \): \( y = \frac{4}{3}x - 2 \).
Therefore, the slope \( m \) is \( \frac{4}{3} \). This is key because parallel lines share the same slope. Transitioning to slope-intercept form gives us a direct view of a line's slope and intercept.
point-slope form
The point-slope form is another way to write the equation of a line. It's particularly useful when you know a point on the line and its slope. The formula is \( y - y_1 = m(x - x_1) \), where
  1. \( (x_1, y_1) \): a point on the line, using its coordinates.
  2. \( m \): the slope of the line.
In our example, we need the line passing through \( (2, -1) \) with a slope of \( \frac{4}{3} \).
  1. Start with the point-slope form: \( y - (-1) = \frac{4}{3}(x - 2) \).
  2. Simplify: \( y + 1 = \frac{4}{3}(x - 2) \).
  3. Distribute \( \frac{4}{3} \): \( y + 1 = \frac{4}{3}x - \frac{8}{3} \).
  4. Subtract \( 1 \): \( y = \frac{4}{3}x - \frac{11}{3} \).
The point-slope form bridges the gap when we know a specific point and the slope, enabling an easy conversion into standard forms of equations.
parallel lines
Understanding parallel lines is crucial. These lines never cross and have identical slopes. The rule is simple: if two lines are parallel, their slopes \( m \) must be equal. Using our exercise:
  1. Given line's slope in slope-intercept form \( y = \frac{4}{3}x - 2 \) is \( \frac{4}{3} \).
  2. The parallel line must also possess a slope of \( \frac{4}{3} \).
Hence, for parallelism, equate the slopes. Moving forward: Identifying slopes ensures we write equations correctly for parallel conditions. This same-slope attribute eliminates guesswork, consolidating our understanding of line relationships. By mastering the characteristic of parallel lines, solving related problems becomes straightforward and less daunting.

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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?

What is the slope of a line that is parallel to the line whose equation is \(y=3 x-8 ?\)

Write an equation of the line satisfying the given conditions. Passing through \((2,3)\) and \((5,9)\)

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.

Sketch the graph of the line satisfying the given conditions. Passing through \((-1,0)\) with slope \(-4\)
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