Problem 11 - 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 \((-2,-3)\) and \((1,4)\)

Step-by-Step Solution

Answer

The equation of the line in standard form is \(7x - 3y = 5\).

Step 1: Calculate the Slope

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the given points \((-2,-3)\) and \((1,4)\) into the formula:\[ m = \frac{4 - (-3)}{1 - (-2)} = \frac{7}{3} \]

Step 2: Use the Point-Slope Form to Find Equation

The point-slope form of a line is given by \[ y - y_1 = m(x - x_1) \]Using the point \((-2, -3)\) and the slope \( m = \frac{7}{3} \), we substitute into the formula:\[ y + 3 = \frac{7}{3}(x + 2) \]

Step 3: Convert the Equation to Standard Form

To convert the equation from point-slope form to standard form, we first simplify the equation and then rearrange the terms. Start by distributing the slope:\[ y + 3 = \frac{7}{3}x + \frac{14}{3} \] Then, clear the fractions by multiplying the entire equation by 3:\[ 3y + 9 = 7x + 14 \] Rearrange to get all terms on one side:\[ 7x - 3y = -9 + 14 \] Finally, simplify the constants:\[ 7x - 3y = 5 \]

Key Concepts

Slope Calculation

Understanding how to calculate the slope is essential for working with line equations. The slope tells you how steep a line is and in what direction it goes (up or down). To find the slope between two points, you use the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For the points \((-2, -3)\) and \(1, 4\), you substitute these values into the formula. Let's do the math:

• Subtract \(-3\) from \(4\): That equals \(7\).
• Subtract \(-2\) from \(1\): That becomes \(3\).
• So the slope \(m = \frac{7}{3}\).

This indicates the line rises \(7\) units for every \(3\) units it goes to the right. A positive slope means the line goes upward!

Point-Slope Form

The point-slope form is a handy way to create the equation of a line when you have a point and the slope. This form looks like:
\[ y - y_1 = m(x - x_1) \]
Here, \(x_1\) and \(y_1\) are coordinates of a point on the line, and \(m\) is the slope. For example, if you use the point \((-2,-3)\) and the slope \(\frac{7}{3}\), it becomes:

• Substitute \(x_1 = -2\) and \(y_1 = -3\) into the formula along with the slope.
• Resulting in \((y + 3 = \frac{7}{3}(x + 2))\).

Point-slope form is very practical because it directly incorporates the slope and a point, making it straightforward to convert to other forms, like standard form.

Standard Form Equation

Once you have the line in point-slope form, you might need to express it in standard form, which is neat and tidy:
\[ Ax + By = C \]
where \(A\), \(B\), and \(C\) are integers, and \(A\) should be positive.

• Starting from \(y + 3 = \frac{7}{3}(x + 2)\): Distribute the slope through \(\frac{7}{3}x + \frac{14}{3}\).
• Clear the fraction by multiplying everything by \(3\).
• That leads to \(3y + 9 = 7x + 14\).
• Rearrange to get all terms on one side: \(7x - 3y = 5\).

This is the standard form equation of the line that passes through \((-2,-3)\) and \(1,4\). It's useful for applications where integer coefficients are preferable.