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Chapter 1: Problem 13

Find equations for these lines: a. Slope 5 and \(y\) intercept \((0,-4)\) b. Slope \(-2\) and contains \((1,3)\) c. Contains \((5,4)\) and is parallel to \(2 x+y=3\)

Short Answer

Expert verified
a. \( y = 5x - 4 \)b. \( y = -2x + 5 \)c. \( y = -2x + 14 \)

Step by step solution

01

Identify the equation form

The general equation of a line in slope-intercept form is given by: \( y = mx + b \). Here, \( m \) is the slope and \( b \) is the y-intercept.
02

Equation for part a

Given the slope \( m = 5 \) and the y-intercept \( (0, -4) \), substitute these values into the slope-intercept form: \( y = 5x - 4 \).
03

Identify point-slope form for part b

The point-slope form of a line is given by: \( y - y_1 = m(x - x_1) \). Here, \( m \) is the slope and \( (x_1, y_1) \) is a point on the line.
04

Equation for part b

Given the slope \( m = -2 \) and the point \( (1, 3) \), substitute these values into the point-slope form: \( y - 3 = -2(x - 1) \). Simplify to the slope-intercept form: \( y = -2x + 5 \).
05

Identify parallel line condition for part c

Lines that are parallel have the same slope. The given line is \( 2x + y = 3 \). First, convert this line to slope-intercept form by solving for \( y \): \( y = -2x + 3 \). The slope of this line is \( -2 \).
06

Use point-slope form for part c

Given the point \( (5, 4) \) and the slope \( -2 \), use the point-slope form: \( y - 4 = -2(x - 5) \). Simplify this to slope-intercept form: \( y = -2x + 14 \).

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Key Concepts

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

Linear Equations
Linear equations describe straight lines on a graph. They have no exponents or curves. In their simplest form, you can write them as: \[ y = mx + b \] Here, \( m \) is the slope, which tells you how steep the line is. True to its name, \( x \) always appears in the equation and \( b \) is the y-intercept, showing where the line crosses the y-axis. Simple examples include:
  • \( y = 3x + 1 \)
  • \( y = 0.5x - 2 \)
  • \( y = -x + 4 \)

Remember, linear equations can always be transformed into the form \( y = mx + b \). This is very useful and forms the foundation for understanding lines on a graph.
Slope-Intercept Form
The slope-intercept form is a specific way to write linear equations. It looks like: \[ y = mx + b \]
  • \( m \) is the slope
  • \( b \) is the y-intercept
The slope tells how much the line rises (or falls) for each unit you move to the right along the x-axis. For example, in the line \( y = 5x - 4 \):
  • The slope \( m = 5 \), meaning the line goes up 5 units for every 1 unit you go right.
  • The y-intercept \( b = -4 \), meaning it crosses the y-axis at -4.
This form is handy for quickly sketching a graph since you know exactly where the line crosses the y-axis and how steep it is.
Point-Slope Form
The point-slope form is often used when you know a point on the line and its slope. The form is: \[ y - y_1 = m(x - x_1)\] Here, \( m \) is the slope, and \( (x_1, y_1) \) is a specific point on the line. For example, if the slope is
  • -2
  • passing through the point (1, 3),
the line equation will be: \( y - 3 = -2 (x - 1)\) This simplifies to the slope-intercept form: \( y = -2x + 5\). The point-slope form is extremely useful when you know a line needs to pass through a specific point.
Parallel Lines
Parallel lines are lines that never intersect. They have the same slope but different y-intercepts. For instance, let's determine the equation of a line parallel to \(2x + y = 3\) that passes through (5, 4). First, we convert \(2x + y = 3\) to slope-intercept form: \(y = -2x + 3\). This tells us the slope is -2. Next, using the point-slope form with the given point (5, 4): \( y - 4 = -2(x - 5) \). Simplify this to: \( y = -2x + 14 \). Hence, the line parallel to \(2x + y = 3\) and passing through (5, 4) is \( y = -2x + 14 \). Understanding the concept of parallel lines helps in solving many geometry and algebra problems efficiently.

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Most popular questions from this chapter

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow 1} \frac{x^{2}+x-2}{x^{2}-1} $$

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MUTATION In a study of mutation in fruit flies, researchers radiate flies with X-rays and determine that the mutation percentage \(M\) increases linearly with the X-ray dosage \(D\), measured in kilo- Roentgens (kR). When a dose of \(D=3 \mathrm{kR}\) is used, the percentage of mutations is \(7.7 \%\), while a dose of \(5 \mathrm{kR}\) results in a \(12.7 \%\) mutation percentage. Express \(M\) as a function of \(D\). What percentage of the flies will mutate even if no radiation is used?
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