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Chapter 4: Problem 17

Find an equation of the line passing through the given points. $$ (9,-0.5) \text { and }(-1,4.5) $$

Short Answer

Expert verified
The equation of the line is \( y = -0.5x + 4 \).

Step by step solution

01

- Identify the given points

The problem provides two points through which the line passes: Point A: \( (9, -0.5) \) Point B: \( (-1, 4.5) \)
02

- Calculate the slope

Use the formula for the slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute \( (x_1, y_1) = (9, -0.5) \) and \( (x_2, y_2) = (-1, 4.5) \): \[ m = \frac{4.5 - (-0.5)}{-1 - 9} = \frac{4.5 + 0.5}{-10} = \frac{5}{-10} = -0.5 \] The slope \( m \) is -0.5.
03

- Use the point-slope form of the equation

Using the point-slope form of a line's equation: \[ y - y_1 = m(x - x_1) \]Substitute \( m = -0.5 \) and the point \( (9, -0.5) \): \[ y - (-0.5) = -0.5(x - 9) \] \[ y + 0.5 = -0.5x + 4.5 \]Subtract 0.5 from both sides to get the standard form: \[ y = -0.5x + 4 \]Thus, the equation of the line is \( y = -0.5x + 4 \).

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Key Concepts

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

slope of a line
The slope of a line tells us how steep the line is. We calculate the slope (\text {m}) as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. If you have two points \text{(x1, y1)} and \text{(x2, y2)}, the formula to find the slope is: \ \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Let's apply this formula using our given points \text {(9, -0.5)} and \text {(-1, 4.5)}:
  • Subtract the y-coordinates: \ (4.5 - (-0.5)) = 4.5 + 0.5 = 5
  • Subtract the x-coordinates: \ (-1 - 9) = -10
  • Divide the results: \ \( m = \frac{5}{-10} = -0.5 \)
The slope of the line passing through the points \text {(9, -0.5)} and \text {(-1, 4.5)} is -0.5. This means for every unit increase in x, y decreases by 0.5 units.
point-slope form
The point-slope form of a linear equation is useful when you know one point on the line and the slope. The formula is: \ \( y - y_1 = m(x - x_1) \).
Here, \text {m} is the slope, and \text {(x1, y1)} is the point.

Let's use the slope we found, -0.5, and one of the points, \text {(9, -0.5)}:
  • Substitute them into the formula: \ \( y - (-0.5) = -0.5 (x - 9) \)
  • Simplify: \ \( y + 0.5 = -0.5x + 4.5 \)
To convert this to the slope-intercept form (\text {y = mx + b}), we subtract 0.5 from both sides:
  • \( y = -0.5x + 4 \)
The equation in point-slope form helps to quickly form an equation when you have the slope and a point.
standard form of a linear equation
The standard form of a linear equation is \text {Ax + By = C}, where A, B, and C are integers. To convert our equation \ \( y = -0.5x + 4 \) to standard form:
  • First, eliminate the decimal by multiplying all terms by 2: \ \( 2y = -x + 8 \).
  • Rearrange to get x and y on one side: \ \( x + 2y = 8 \)
Now, the equation is in standard form.

It's useful for many applications, such as finding intercepts and solving systems of equations.

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