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Chapter 1: Problem 9

Find an equation of the given line. Slope is \(2 ;(1,-2)\) on line.

Short Answer

Expert verified
y = 2x - 4

Step by step solution

01

Identify the slope

The slope of the line is given as 2. This means that for every unit increase in x, y increases by 2 units.
02

Use point-slope form

The point-slope form of the equation of a line is y - y1 = m(x - x1);, where m is the slope and (x1, y1) is a point on the line. Substitute m = 2, x1 = 1, and y1 = -2 into the equation.
03

Substitute the given values

Plugging the values into the point-slope equation we get: y + 2 = 2(x - 1).
04

Simplify the equation

Distribute the 2 on the right-hand side: y + 2 = 2x - 2. Now, isolate y to get the equation in slope-intercept form:
05

Rewrite in slope-intercept form

Subtract 2 from both sides: y = 2x - 4. This is the equation of the line in slope-intercept form.

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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 is a measure of its steepness. It indicates how much the y-coordinate increases or decreases as the x-coordinate increases by 1 unit. Mathematically, the slope (m) is calculated by the formula:
\[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \]
where
  • (x1, y1) and
  • (x2, y2)
are any two distinct points on the line. A positive slope means the line rises as we move from left to right, and a negative slope means it falls. For example, if a line has a slope of 2, like in our exercise, this means it rises 2 units for every 1 unit of horizontal movement to the right.
point-slope equation
The point-slope form of a line's equation is particularly useful when you know one point on the line and the slope. The general form is:
\[ y - y_1 = m(x - x_1) \]
Here,
  • m is the slope, and
  • (x1, y1) is a specific point on the line.
To derive a line's equation using this form, substitute the given point and slope into the formula. For our exercise, the slope m is 2, and the point (x1, y1) is (1, -2). Substituting these values gives:
\[ y - (-2) = 2(x - 1) \]
This simplifies to:
\[ y + 2 = 2(x - 1) \]
slope-intercept form
The slope-intercept form of a line's equation is one of the most straightforward and widely used formats. It is written as:
\[ y = mx + b \]
Here,
  • m is the slope, and
  • b is the y-intercept (the y-coordinate where the line crosses the y-axis).
To convert a point-slope form equation into slope-intercept form, you typically isolate y. Starting from the point-slope equation in the exercise:
\[ y + 2 = 2(x - 1) \]
Distributing 2 on the right side results in:
\[ y + 2 = 2x - 2 \]
Lastly, subtracting 2 from both sides to isolate y gives:
\[ y = 2x - 4 \]
Therefore, the line's equation in slope-intercept form is:
\[ y = 2x - 4 \]

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