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Chapter 3: Problem 32

(a) rewrite the equation in slope-intercept form. (b) identify the slope. (c) identify the \(y\)-intercept. Write the ordered pair, not just the \(y\)-coordinate. (d) find the \(x\)-intercept. Write the ordered pair, not just the \(x\)-coordinate. $$ 8 x-9 y=-72 $$

Short Answer

Expert verified
Slope-intercept form: \(y = \frac{8}{9} x + 8\). Slope: \(\frac{8}{9}\). Y-intercept: \((0, 8)\). X-intercept: \((-9, 0)\).

Step by step solution

01

- Rewrite in Slope-Intercept Form

Slope-intercept form is given by: \(y = mx + b\). Start by isolating \(y\). Given the equation: \[8x - 9y = -72\] First, subtract \(8x\) from both sides:\[-9y = -8x - 72\]. Next, divide every term by \(-9\):\[y = \frac{8}{9}x + 8\].
02

- Identify the Slope

In the slope-intercept form equation \(y = mx + b\), the slope \(m\) is the coefficient of \(x\). Here, \(m = \frac{8}{9}\).
03

- Identify the Y-intercept

In the slope-intercept form equation \(y = mx + b\), the \(y\)-intercept \(b\) is the constant term. Here, \(b = 8\). The \(y\)-intercept is the point where \(x = 0\), which gives the ordered pair: \((0, 8)\).
04

- Find the X-intercept

The \(x\)-intercept is the point where \(y = 0\). Set \(y = 0\) in the slope-intercept equation:\[0 = \frac{8}{9}x + 8\]. Solve for \(x\) by first subtracting 8 from both sides:\[-8 = \frac{8}{9}x\]. Then, multiply both sides by \(\frac{9}{8}\):\[x = -9\]. The \(x\)-intercept is the point \((-9, 0)\).

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Key Concepts

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

linear equations
A linear equation is a type of equation that represents a straight line when graphed on a coordinate plane. The standard form of a linear equation is written as: \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are constants, with \(x\) and \(y\) being variables. The purpose of converting a linear equation into slope-intercept form is to easily identify the slope and the y-intercept. Linear equations are essential in mathematics because they model relationships where one variable depends on another in a direct, proportional way. They are used in various fields such as physics, engineering, and economics to represent real-world problems efficiently.
slope
The slope of a line, often represented by the letter \(m\), measures the steepness and direction of the line. You can find the slope by taking the coefficient of \(x\) in the slope-intercept form equation, which is \(y = mx + b\). In our example, once we converted the equation into slope-intercept form, the equation became: \(y = \frac{8}{9}x + 8\), so the slope \(m\) is \(\frac{8}{9}\). The slope tells us how much \(y\) changes for a unit change in \(x\):
  • If the slope is positive, the line rises as it moves from left to right.
  • If the slope is negative, the line falls as it moves from left to right.
This concept is crucial in understanding how variables interact in various scenarios.
y-intercept
The \(y\)-intercept is the point where the line crosses the \(y\)-axis. In slope-intercept form, \(y = mx + b\), the \(y\)-intercept is represented by the constant term \(b\). For our equation, \(y = \frac{8}{9}x + 8\), the \(y\)-intercept is 8. This means that when \(x = 0\), \(y = 8\). Therefore, the ordered pair representing the \(y\)-intercept is \((0, 8)\). This point is significant because it provides a starting point and is necessary for graphing the line accurately and understanding the behavior of the function when \(x\) is zero.
x-intercept
The \(x\)-intercept is where the line crosses the \(x\)-axis. It is found by setting \(y = 0\) in the slope-intercept form equation and solving for \(x\). For the equation \(y = \frac{8}{9}x + 8\), set \(y = 0\): \(0 = \frac{8}{9}x + 8\). Solving for \(x\) involves:
  • First, subtract 8 from both sides to get \(-8 = \frac{8}{9}x\).
  • Then, multiply both sides by \(\frac{9}{8}\) to isolate \(x\).
This gives \(x = -9\). Therefore, the \(x\)-intercept is the point \((-9, 0)\). This point helps us understand where the line intersects the \(x\)-axis, providing important information about the function's behavior with respect to \(x\).

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Most popular questions from this chapter

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((1,5) ;(3,13)\)

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

Use the slope formula to find the slope of the line that passes through the points. \(\left(-8, \frac{1}{4}\right) ;\left(16, \frac{1}{2}\right)\)

(a) write the equation of the vertical line that passes through the point. (b) graph the equation. \((-2,-1)\)

For exercises 37-66, use the slope formula to find the slope of the line that passes through the points. \((1,9) ;(3,14)\)
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