Problem 21 - Calculus for Biology and Medicine

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope \(1 / 2\) and \(y\) -intercept \((0,2)\)

Step-by-Step Solution

Answer

The line equation in standard form is \( x - 2y = -4 \).

Step 1: Understanding the Line Equation

The problem asks us to determine the equation of a line in standard form. The slope-intercept form of a line is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Given in the problem, the slope \( m \) is \( \frac{1}{2} \) and the \( y \)-intercept is \( 2 \), represented as the point \( (0, 2) \).

Step 2: Write Equation in Slope-Intercept Form

Insert the given values of slope and \( y \)-intercept into the slope-intercept equation: \( y = \frac{1}{2}x + 2 \). This is the equation of the line in slope-intercept form.

Step 3: Convert to Standard Form

The standard form of a line is given by \( Ax + By = C \). To convert from slope-intercept to standard form, start by clearing the fraction. Multiply every term by 2 to get rid of the denominator: \( 2y = x + 4 \). Then rearrange terms to get \( x - 2y = -4 \).

Step 4: Verify the Equation

To ensure the accuracy, substitute \( x = 0 \) into the standard form equation \( x - 2y = -4 \) to check if \( y = 2 \). The equation yields \( -2y = -4 \) giving \( y = 2 \), which is correct as per the \( y \)-intercept given. Hence, the equation \( x - 2y = -4 \) is verified.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most important equations to know when dealing with lines in mathematics. It is expressed as \( y = mx + b \). Here, \( m \) represents the slope of the line, which essentially tells us how steep the line is. The slope describes the rate of change in \( y \) for every unit increase in \( x \).

• For positives slopes, the line ascends, meaning it goes up from left to right.
• For negative slopes, the line descends, moving downward from left to right.

The second part, \( b \), is the \( y \)-intercept, which is the point on the graph where the line crosses the \( y \)-axis. This means, if you set \( x = 0 \), \( y \) will be \( b \). To utilize this form, you only need two components: the slope \( m \) and the \( y \)-intercept \( b \). By inserting these values in the equation, you get a complete picture of the line on a graph.

Standard Form

Standard form is another way to express the equation of a line. It is written as \( Ax + By = C \) where \( A \), \( B \), and \( C \) are integers and \( A \) should be a positive number. This form is particularly useful when dealing with simultaneous equations or when integrating with other mathematical concepts.

• One advantage is that it easily shows the interaction between \( x \) and \( y \) because both variables are on the same side of the equation.
• Remember that to convert from slope-intercept form to standard form, follow these steps:
• First, clear any fractions by multiplying through by the denominator.
• Rearrange the equation so \( x \) and \( y \) are on the left side and the constant is on the right.

Standard form also comes in handy when you want to graph a line using the intercepts. Simply substitute 0 for \( x \) to find the \( y \)-intercept, and 0 for \( y \) to find the \( x \)-intercept.

Y-Intercept

The \( y \)-intercept is a crucial component in understanding the behavior of a line on a graph. It is simply the point where the line crosses the \( y \)-axis. Mathematically, this happens when the value of \( x \) is zero. In the slope-intercept form \( y = mx + b \), the \( y \)-intercept is represented by \( b \).

• This is the starting point of the line when you are plotting from the \( y \)-axis.
• The \( y \)-intercept provides initial information about the position of the line without needing to plot several points.

Knowing the \( y \)-intercept helps in sketching the graph of a line, because along with the slope, it allows you to visualize where the line will be positioned even before you draw. For example, with a \( y \)-intercept of \( 2 \), as given, you know instantly that the line will cross the vertical axis at \( (0, 2) \).