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Chapter 3: Problem 9

A line with the given slope passes through the given point. Write the equation of the line in slope-intercept form. slope \(=-\frac{2}{3} ;(3,-6)\)

Short Answer

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

Step by step solution

01

- Recall the slope-intercept form

The slope-intercept form of a line's equation is given by the formula: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
02

- Substitute the given slope

The given slope is \(-\frac{2}{3}\). Substitute this value into the slope-intercept form equation: \(y = -\frac{2}{3}x + b\).
03

- Substitute the given point

The given point is \((3, -6)\). Substitute \(x = 3\) and \(y = -6\) into the equation: \(-6 = -\frac{2}{3}(3) + b\).
04

- Solve for the y-intercept

Simplify the equation to solve for \(b\). Starting with: \(-6 = -2 + b\). Add 2 to both sides to isolate \(b\): \(-6 + 2 = b\). Thus, \(b = -4\).
05

- Write the final equation

Substitute \(b = -4\) back into the slope-intercept form equation: \(y = -\frac{2}{3}x - 4\).

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Key Concepts

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

Linear Equations
Understanding linear equations is fundamental to learning algebra. A linear equation is an equation that models a straight line. It is typically written in the form:
  • Standard Form: \(Ax + By = C\)
  • Slope-Intercept Form: \(y = mx + b\)
  • Point-Slope Form: \(y - y_1 = m(x - x_1)\)
Each form has its use, but the slope-intercept form is particularly useful for quickly identifying a line's slope and y-intercept.
Slope of a Line
The slope of a line measures its steepness and direction. It is represented by \(m\) in the slope-intercept form \(y = mx + b\). Slope is calculated as the 'rise' over the 'run,' or the change in \(y\) over the change in \(x\). For the given slope \(-\frac{2}{3}\), it means that for every 3 units you move to the right (positive x-direction), the line falls by 2 units (negative y-direction).
  • Slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Understanding the slope helps in predicting how the line behaves and intersects other points.
Y-Intercept
The y-intercept is where the line crosses the y-axis. It is represented by \(b\) in the slope-intercept form \(y = mx + b\). To find it, we substitute the values of a point on the line along with the slope into the equation and solve for \(b\). In this exercise, after substituting \(x = 3\) and \(y = -6\) into the equation, we calculated \(b = -4\). Thus, the y-intercept is -4, which means the line crosses the y-axis at the point \( (0, -4) \). Identifying the y-intercept is key in graphing the line accurately.
Point-Slope Form
The point-slope form of a line's equation is another useful way to represent linear equations. It is: \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a specific point on the line. This form is particularly useful when you know one point on the line and the slope, but not the y-intercept. In this exercise:
  • Given point: (3, -6)
  • Given slope: -\(\frac{2}{3}\)
We can use the point-slope form to derive the slope-intercept form by simplifying it to determine \(b\). Essentially, these forms connect different pieces of information about the line, making it easier to transition between them during problem-solving.

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Most popular questions from this chapter

Use the slope formula to find the slope of the line that passes through the points. \((-1,5) ;(-6,-13)\)

Balanced Rock in Arches National Park is \(55 \mathrm{ft}\) tall and weighs 3500 tons. Find its height in meters. Round to the nearest tenth. \((1 \mathrm{~m} \approx 3.2808 \mathrm{ft}\).) (Source: www.desertusa .com)

(a) find three solutions of the equation. (b) graph the equation. \(y=-x\)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(3 x+10 y=60\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(0, \frac{4}{5}\right) ;\left(\frac{1}{5}, 0\right)\)
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