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Chapter 5: Problem 46

Determine the slope of the line from its equation. $$5 x-6 y-7=0$$

Short Answer

Expert verified
The slope is \( \frac{5}{6} \).

Step by step solution

01

- Rewrite the equation in slope-intercept form

To find the slope, rearrange the given equation into the slope-intercept form, which is \text{slope-intercept form} \text{is} y = mx + b. The given equation is :$$ 5x - 6y - 7 = 0 $$.
02

- Isolate the \(y\)-term

Rewrite the equation to isolate the y-term on one side. Starting with: $$ 5x - 6y - 7 = 0,$$ Add 7 to both sides to get: $$ 5x - 6y = 7.$$
03

- Solve for \(y\)

Now, get y by itself. Subtract 5x from both sides: $$ -6y = -5x + 7.$$ Then divide everything by -6: $$ y = \frac{5}{6}x - \frac{7}{6}.$$ The equation is now in the form $$ y = mx + b$$.
04

- Identify the slope

The slope-intercept form is $$ y = mx + b$$ , where m is the slope. From the equation $$ y = \frac{5}{6}x - \frac{7}{6}$$, identify the slope m as \( \frac{5}{6} \).

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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 a fundamental concept in algebra. It is expressed as \(y = mx + b\). This form allows you to easily identify the slope \(m\) and the y-intercept \(b\) of a line.

In the given exercise, our task is to convert the equation \(5x - 6y - 7 = 0\) into this form. The slope-intercept form makes it straightforward to determine the slope by turning the equation into a format where the coefficient of \(x\) directly represents the slope.

To do this:
  • First, isolate the \(y\)-term on one side of the equation.
  • Then, rearrange the remaining terms so the equation reads \(y = mx + b\).
In our case, once the equation is rearranged to \(y = \frac{5}{6}x - \frac{7}{6}\), we find the slope \(m\) is \frac{5}{6}\.
Solving Linear Equations
Solving linear equations is a crucial skill in algebra. It involves finding the value of the variable that makes the equation true.

In the step-by-step solution, solving the linear equation \(5x - 6y - 7 = 0\) involves several steps:
  • Add or subtract terms to both sides of the equation move all terms involving \(x\) and constants to one side, and isolate the \(y\)-term on one side. For our example, we add 7 to both sides getting \(5x - 6y = 7\).

  • Then, solve for the \(y\)-term by moving \(x\)-terms to the other side and dividing through by the coefficient of \(y\). In this case, divide by -6 to get \(y = \frac{5}{6}x - \frac{7}{6} \).
Using these steps systematically helps you solve the equation and rewrite it in slope-intercept form.
Rearranging Equations
Rearranging equations is essential to simplifying and solving them. This often includes moving terms from one side to the other, factoring, combining like terms, and isolating variables. In algebra, these manipulations help change the form of an equation for easier handling.

In our equation \(5x - 6y - 7 = 0\), we needed to isolate the \(y\)-term to rewrite the equation in slope-intercept form:
  • First, move any constants to the other side of the equation \(5x - 6y = 7\)
  • Then, rearrange to isolate \(y\) on one side \(-6y = -5x + 7\).
  • Finally, divide every term by the coefficient of \(y\) (in this case, -6) to complete the rearrangement \(y = \frac{5}{6}x - \frac{7}{6}\).
Rearranging the equation this way prepares it for easy identification of the slope \ \frac{5}{6} \ and enhances our understanding of the line's properties.

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Most popular questions from this chapter

This exercise discusses the relationship between the slopes of perpendicular lines. (a) Sketch the graphs of \(y=2 x+4\) and \(y=-\frac{1}{2} x+4\) on the same coordinate system. (b) On the basis of your graph, does it appear that these lines are perpendicular? (c) What is the relationship between the slopes of these two lines? (d) It is a fact that the slopes of perpendicular lines are negative reciprocals of each other (provided that neither of the lines is vertical). What is the slope of a line perpendicular to the line whose equation is \(y=\frac{2}{5} x+7 ?\)

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?

Sketch the graph of the line satisfying the given conditions. Passing through \((-1,-5)\) and whose slope is undefined

A seamstress can hem a skirt in 10 minutes and make the cuffs on a pair of pants in 15 minutes. On a certain day she has allotted 300 minutes to doing these tasks. (a) Let \(s\) represent the number of skirts she hems and \(p\) represent the number of pairs of pants she cuffs. Write an equation that reflects the given situation. (b) Sketch the graph of this relationship. Be sure to label the coordinate axes clearly. (c) If she hems 18 skirts, use the equation you obtained in part (a) to find the number of pairs of pants she cuffs.

Sketch the graph of the line satisfying the given conditions. Passing through \((1,-2)\) with slope \(\frac{-2}{3}\)
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