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Chapter 4: Problem 498

Find the equation of each line. Write the equation in slope-intercept form. Perpendicular to the line \(x-2 y=5\), containing point (-2,2)

Short Answer

Expert verified
The equation of the line is \( y = -2x - 2 \).

Step by step solution

01

- Rewrite given line equation in slope-intercept form

Rewrite the given equation \( x - 2y = 5 \) in slope-intercept form \( y = mx + b \): Start by isolating \( y \): \[ x - 2y = 5 \] Subtract \( x \) from both sides: \[ -2y = -x + 5 \] Divide both sides by -2: \[ y = \frac{1}{2}x - \frac{5}{2} \] So, the slope \( m \) of the given line is \( \frac{1}{2} \).
02

- Find slope of the perpendicular line

Perpendicular lines have slopes that are negative reciprocals of each other. The slope of the given line is \( \frac{1}{2} \), so the slope of the perpendicular line \( m_{perpendicular} \) is: \[ m_{perpendicular} = - \frac{1}{\frac{1}{2}} = -2 \]
03

- Use point-slope form to find the equation

Use the point-slope form of the equation \( y - y_1 = m(x - x_1) \) to find the equation of the perpendicular line that passes through the point \( (-2, 2) \).Plug in \( m = -2 \), \( x_1 = -2 \), \( y_1 = 2 \):\[ y - 2 = -2(x + 2) \]Simplify to find the equation:
04

- Simplify to slope-intercept form

Expand and simplify to get to slope-intercept form \( y = mx + b \):\[ y - 2 = -2x - 4 \]Add 2 to both sides:\[ y = -2x - 2 \]

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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 models a straight line. Its general form is usually written as ax + by + c = 0 or y = mx + b. Here, 'a', 'b', and 'c' are constants, and 'x' and 'y' are variables. The term 'm' represents the slope, and 'b' represents the y-intercept, the point where the line crosses the y-axis.

In simpler terms, a linear equation helps you understand how y changes with respect to x in a predictable way.

If you change the value of x, you can find a corresponding value of y. This relationship is always a straight-line when graphed.
  • The graph is always a straight line
  • The slope (m) measures steepness
  • Y-intercept (b) tells where it crosses y-axis
slope
The slope is a measure of the steepness of a line. Think of it as 'rise over run', which means how much a line goes up (or down) as it goes across. Mathematically, the slope 'm' between two points (x1, y1) and (x2, y2) on a line is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

The slope can be positive, negative, zero, or undefined:

  • Positive slope: The line rises as it goes from left to right.
  • Negative slope: The line falls as it goes from left to right.
  • Zero slope: The line is horizontal, no change in y.
  • Undefined slope: The line is vertical, no change in x.

Understanding slope is crucial. In our exercise, the original line's slope was \( \frac{1}{2} \), leading us to find the perpendicular slope as -2.
perpendicular lines
Perpendicular lines intersect at right angles (90 degrees). A significant property of these lines is that their slopes are negative reciprocals. In easier terms, if you have a slope 'm', a line perpendicular to it will have a slope of \( -\frac{1}{m} \).

For example, if the slope of the given line is \( \frac{1}{2} \), the perpendicular line's slope is -2. Always changing the sign and flipping the fraction.

To find the equation of a perpendicular line in our exercise, this property helped us determine the new slope.
point-slope form
The point-slope form of a line is useful when you know a point on the line and the slope. It's written as:

\[ y - y_1 = m(x - x_1) \]

Here, (x1, y1) are coordinates of a point on the line, and 'm' is the slope.

This form is particularly helpful for quickly finding the equation given enough information. In our exercise, we used the point-slope form to find the equation of the perpendicular line passing through (-2,2). Plugging in gives:

\[ y - 2 = -2(x + 2) \]
Simplify the equation to get it into slope-intercept form as needed.
Understanding point-slope form bridges the gap to slope-intercept form easily.

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