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Chapter 11: Problem 222

Find the slope, the \(\mathrm{y}\) -intercept, and the \(\mathrm{x}\) -intercept of the equation \(2 x-3 y-18=0\)

Short Answer

Expert verified
The slope of the equation is \(m = \frac{2}{3}\), the y-intercept is \(b = -6\), and the x-intercept is \(x = 9\).

Step by step solution

01

Rewrite the equation in slope-intercept form (y = mx + b)

To write the given equation in slope-intercept form (y = mx + b), we need to first move the 3y term to the other side and then divide each term by 3: \(2x - 3y - 18 = 0\) Add 3y to both sides and get: \(2x - 18 = 3y\) Now divide each term by 3: \(y = \frac{2}{3}x - 6\)
02

Determine the slope and y-intercept

In the equation we obtained in Step 1, the slope (m) is the coefficient of the x term, and the y-intercept (b) is the constant term. From the equation: \(y = \frac{2}{3}x - 6\) The slope (m) is: \(m = \frac{2}{3}\) The y-intercept (b) is: \(b = -6\)
03

Determine the x-intercept

To find the x-intercept, we set the value of y to 0 and solve for x: \(\frac{2}{3}x - 6 = 0\) Add 6 to both sides: \(\frac{2}{3}x = 6\) Now multiply both sides by \(\frac{3}{2}\) to solve for x: \(x = \frac{3}{2} \cdot 6\) \(x = 9\) The x-intercept is 9. So, the slope of the equation is \(\frac{2}{3}\), the y-intercept is -6, and the x-intercept is 9.

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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 an equation that creates a straight line when graphed on a coordinate plane. These equations are generally presented in different forms, such as the standard form, slope-intercept form, or point-slope form.
In this context, we are interested in the slope-intercept form, which is represented as \(y = mx + b\). Here:
  • \(m\) is the slope of the line.
  • \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
Linear equations are very important in mathematics and are used to model real-world situations that show a constant rate of change.
One of the main benefits of understanding linear equations is their simplicity and reliability in predicting and explaining relationships between variables.
X-Intercept
The x-intercept of a line is a point where the line crosses the x-axis. This is found by setting the y-value of the linear equation to zero and solving for x.
In simple terms, it tells us the value of x when y is zero, helping us understand where the line "hits" the horizontal axis on a graph.
For example, in our equation \(\frac{2}{3}x - 6 = 0\), we calculated the x-intercept by:
  • Adding 6 to both sides, resulting in \(\frac{2}{3}x = 6\).
  • Multiplying both sides by \(\frac{3}{2}\), leading us to \(x = 9\).
Thus, the x-intercept is 9. Understanding this concept is crucial for graphing and interpreting linear equations in a coordinate plane.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. This is determined by setting the x-value of the linear equation to zero and solving for y, but in the slope-intercept form \(y = mx + b\), the y-intercept is directly given by the constant term \(b\).
This intercept represents the value of y when x is zero.
In the equation \(y = \frac{2}{3}x - 6\), we can see that the y-intercept is -6, meaning the line crosses the y-axis at the point (0, -6).
  • This information is useful because it gives us a starting point for graphing the line.
  • Knowing the y-intercept helps in understanding the initial value or starting point in real-world scenarios modeled by linear equations.
Slope of a Line
The slope of a line is a number that describes both the direction and the steepness of the line. It is represented by the letter \(m\) in the slope-intercept equation \(y = mx + b\).
The slope is defined as the ratio of the change in y to the change in x between two points on the line.
In our example equation, \(y = \frac{2}{3}x - 6\), the slope is \(\frac{2}{3}\). This slope tells us that for every 3 units the line moves horizontally (right), it moves 2 units vertically (up).
  • A positive slope indicates that the line goes upwards as it moves from left to right.
  • A negative slope would mean the line slopes downwards.
  • If the slope is zero, the line would be horizontal, indicating no vertical change as it moves.
Understanding slope is key to interpreting and predicting trends from linear models.

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Most popular questions from this chapter

The equation \(\mathrm{F}=9 / 5 \mathrm{C}+32\) relates the \(\mathrm{Fahrenheit}\) and centigrade temperature scales. What do the numbers \(9 / 5\) and 32 represent?

Graph the constant function \(2 \mathrm{y}=4\).

The slope and one point of a line are given. Is the Y-intercept positive or negative? (a) \(\mathrm{m}=22 / 7,(1, \pi)\) b) \(\mathrm{m}=\sqrt{2},(1,1.414)\)

a) Find the zeros of the function \(\mathrm{f}\) if \(\mathrm{f}(\mathrm{x})=3 \mathrm{x}-5\). b) sketch the graph of the equation \(\mathrm{y}=3 \mathrm{x}-5\).

Find the equation for the line passing through \((3,5)\) and \((-1,2)\).
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