Problem 13 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

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 \(2 x+y=3\)

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: Equation for part a

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

Step 3: 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.

Step 4: 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: \( y = -2x + 5 \).

Step 5: 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 \( -2 \).

Step 6: 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: \( y = -2x + 14 \).

Key Concepts

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.