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Chapter 3: Problem 28

A line passes through the given points. (a) Find the slope of the line. (b) Write the equation of the line in slope-intercept form. $$ (-6,30) ;(-14,24) $$

Short Answer

Expert verified
The slope is \( \frac{3}{4} \). The equation of the line is \( y = \frac{3}{4}x + 34.5 \).

Step by step solution

01

Title - Find the Slope

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the given points \((-6, 30)\) and \((-14, 24)\) into the formula: \[ m = \frac{24 - 30}{-14 - (-6)} = \frac{-6}{-8} = \frac{3}{4} \] Therefore, the slope of the line is \frac{3}{4}.\
02

Title - Use Slope-Intercept Form

The slope-intercept form of a line is given by the equation: \[ y = mx + b \] We already have the slope \(m = 3/4\). Use one of the given points to find the y-intercept \(b\). Let's use \((-6, 30)\). Substitute the values into the equation: \[ 30 = \frac{3}{4}(-6) + b \] Simplify and solve for \(b\): \[ 30 = -\frac{18}{4} + b 30 = -4.5 + b b = 30 + 4.5 b = 34.5 \] Therefore, the y-intercept is \34.5\.
03

Title - Write the Final Equation

Substitute the slope \(m = 3/4\) and the y-intercept \(b = 34.5\) back into the slope-intercept form equation: \[ y = \frac{3}{4}x + 34.5 \] This is the equation of the line.

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Key Concepts

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

Finding the Slope
To determine the slope of a line passing through two points, you can use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. The slope (m) measures the steepness or inclination of the line between two points, \((x_1, y_1)\) and \((x_2, y_2)\).

  • Identify the coordinates of the two points. For instance, the points given are \((-6, 30)\) and \((-14, 24)\).
  • Substitute these coordinates into the slope formula:

\[ m = \frac{24 - 30}{-14 - (-6)} = \frac{-6}{-8} = \frac{3}{4} \]

This indicates that the slope of the line is \(\frac{3}{4}\). The slope tells us that as you move 4 units horizontally, the line moves up 3 units vertically.
Equation of a Line
The equation of a line can be written in slope-intercept form: \[ y = mx + b \]. Here, \(m\) represents the slope and \(b\) is the y-intercept.

  • We already have the slope, m, which is \(\frac{3}{4}\).
  • Next, we need to find the y-intercept \((b)\).
  • To find \(b\), substitute \(m\) and any of the given points into the equation. Let's use the point \((-6, 30)\):

\[ 30 = \frac{3}{4}(-6) + b \]

Now, solve for \(b\):
\[ 30 = -\frac{18}{4} + b \], which simplifies to \[ 30 = -4.5 + b \]. Add 4.5 to both sides:
\[ b = 30 + 4.5 \]
\[ b = 34.5 \]

Therefore, the y-intercept is 34.5.
Y-Intercept
The y-intercept \((b)\) is where the line crosses the y-axis.
It can be found by setting \(x = 0\) in the line equation and solving for \(y\).
In the slope-intercept form, \(y = mx + b\), \(b\) is already isolated and represents the y-intercept directly.

  • Based on our earlier calculation, we obtained the y-intercept as 34.5.
  • This value indicates that the line passes through the point \((0, 34.5)\) on the y-axis.


Finally, combining both the slope and the y-intercept, we can write the final equation of the line as:
\[ y = \frac{3}{4}x + 34.5 \] This is the completed equation in slope-intercept form, which shows both the slope and the y-intercept of the line.

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Most popular questions from this chapter

Use a graphing calculator to graph each equation. Choose a window that shows the \(x\)-intercept and \(y\)-intercept. Sketch the graph; describe the window. \(y=\frac{1}{2} x-8\)

Use the slope formula to find the slope of the line that passes through the points. \((4 \mathrm{hr}, \$ 36) ;(6 \mathrm{hr}, \$ 54)\)

Explain why the slope of a horizontal line is always zero.

(a) use a graphing calculator to graph \(y=\frac{3}{4} x-5\). Sketch the graph; describe the window. (b) for the given value of \(x\), use the Value command to find the value of \(y\). If it is necessary to change the window, describe the changed window. \(x=20\)

A high-speed Shinkansen train in Japan travels at a speed of \(\frac{270 \mathrm{~km}}{1 \mathrm{hr}}\) for \(18 \mathrm{~min}\). Find the distance it travels.
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