Problem 33 - 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 passing through \((5,-1)\) and perpendicular to the line passing through \((-2,1)\) and \((1,-2)\)

Step-by-Step Solution

Answer

The equation in standard form is \(x - y = 6\).

Step 1: Find slope of the given line

To determine the slope of the line passing through the points (-2,1) and (1,-2), we'll use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substituting the given points: \[ m = \frac{-2 - 1}{1 - (-2)} = \frac{-3}{3} = -1 \]So, the slope of the given line is -1.

Step 2: Find slope of the perpendicular line

Since perpendicular lines have slopes that are negative reciprocals of each other, we'll find the negative reciprocal of -1. The negative reciprocal of -1 is 1. Hence, the slope of the line we're looking for is 1.

Step 3: Write the equation in point-slope form

Using the point (5,-1) and the slope 1, substitute into the point-slope equation of a line: \[ y - y_1 = m(x - x_1) \] Substitute \( x_1 = 5, y_1 = -1, m = 1 \): \[ y - (-1) = 1(x - 5) \] Simplifying, we get: \[ y + 1 = x - 5 \]

Step 4: Convert to standard form

To convert the equation \( y + 1 = x - 5 \) to standard form, we rearrange it to make one side equal to zero and the coefficients integers: \[ y + 1 = x - 5 \] Subtract \(x\) from both sides: \[ -x + y + 1 = -5 \] Subtract 1 from both sides: \[ -x + y = -6 \] To follow the standard form where the coefficient of x is positive, multiply through by -1: \[ x - y = 6 \]

Key Concepts

Equation of a Line

Understanding how to find the equation of a line is a fundamental concept in coordinate geometry. An equation of a line expresses the relationship between the x and y coordinates of any point on the line. The general form of a line equation is usually expressed as either

• Slope-intercept form: \( y = mx + b \)
• Point-slope form: \( y - y_1 = m(x - x_1) \)
• Standard form: \( Ax + By = C \)

where \(m\) represents the slope of the line, \(b\) is the y-intercept, and \(A\), \(B\), and \(C\) are integers in the standard form. To solve problems, you often start with either the point-slope or slope-intercept form and rearrange terms to achieve the desired equation format. Converting between forms requires a strong grasp of algebraic manipulation.

Point-Slope Form

The Point-Slope Form is a useful way to express the equation of a line when you know a point on the line and its slope. It's given by the formula \( y - y_1 = m(x - x_1) \). Here, the \((x_1, y_1)\) represents a specific point on the line, and \(m\) represents the slope.

This form is especially handy when you are given a problem involving a specific point and are asked to know or find the slope. In our exercise, we used the point \((5, -1)\) and a slope of 1 to establish the point-slope form, resulting in \( y + 1 = x - 5 \).

From this step, converting to other forms like the standard form becomes more straightforward. The flexibility and straightforwardness in the point-slope form make it a favorite starting point for many line equations.

Standard Form

The Standard Form of a line's equation is represented as \( Ax + By = C \), where \(A\), \(B\), and \(C\) are integers, and typically \(A\) should be positive. The standard form is particularly useful in scenarios where you want to analyze or assess properties of the line based on whole number coefficients, as it can provide a clearer picture when comparing or combining equations.

To convert from another form, like the point-slope or slope-intercept form, to the standard form, rearrange terms so that x and y variables are on one side of the equation and constants are on another. In our exercise, we transformed the equation \( y + 1 = x - 5 \) into the standard form \( x - y = 6 \). This involved basic algebraic steps, such as shifting terms and ensuring the coefficients meet the integer and positive criteria.

Perpendicular Lines

In coordinate geometry, perpendicular lines are lines that intersect at a right angle (90 degrees). A key characteristic of perpendicular lines is their slopes are negative reciprocals of each other. For example, if one line has a slope of \( m \), a line perpendicular to it will have a slope of \(-\frac{1}{m}\). In our case, a line with a slope of \(-1\) will be perpendicular to a line with a slope of 1 because \(-1\) and 1 are negative reciprocals.

Understanding perpendicular relationships helps in solving problems where you're asked to find the equation of a line with a specific relationship to another. Given this property, once you know the slope of a line, you can quickly find the slope of any line perpendicular to it. This is a common technique used in geometry to construct various line equations or to solve for unknown line parameters.