Problem 15 - The Calculus with Analytic Geometry

Question

$$ \text { Find an equation of the line through the points }(1,3) \text { and }(2,-2) \text {, and put the equation in the intercept form. } $$

Step-by-Step Solution

Answer

The equation of the line is \(5x + y = 8\).

Step 1: Find the slope

Use the slope formula to find the slope of the line passing through the points \(1, 3\) and \(2, -2\). The slope formula is \(\frac{y_2 - y_1}{x_2 - x_1}\). \(\)Substitute the given points into the slope formula: \[\frac{-2 - 3}{2 - 1} = \frac{-5}{1} = -5\]

Step 2: Use the point-slope form

With the slope found in Step 1, use the point-slope form of the equation of a line, which is \(y - y_1 = m(x - x_1)\). Choose the point (1,3) for \(x_1, y_1\) and \( -5 \) for the slope \(m\)\(y - 3 = -5(x - 1)\)

Step 3: Simplify the equation

Distribute and simplify the equation obtained in Step 2. \(y - 3 = -5x + 5 \)\(y = -5x + 8\)

Step 4: Convert to intercept form

The standard intercept form equation is \(Ax + By = C\). Rearrange \(y = -5x + 8\) to fit this form. Add \(5x\) to both sides: \(\5x + y = 8\) n This is the equation in intercept form.

Key Concepts

slope formula

The slope formula is essential for finding the steepness or incline of a line. It helps us understand how much a line rises or falls over a certain distance. The formula for slope is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here,

• \( (x_1, y_1) \) and \((x_2, y_2)\) are two points on the line.
• \( m \) is the slope of the line.

Using this formula, we can find the slope between any two points. For example, given the points (1, 3) and (2, -2), we substitute them into the slope formula like this: \[ m = \frac{-2 - 3}{2 - 1} = \frac{-5}{1} = -5 \] In this case, the slope \( m \) is -5, showing that the line falls 5 units for every 1 unit it moves to the right. Understanding the slope is the first step in finding the equation of a line.

point-slope form

The point-slope form of a line's equation is very useful when you know one point on the line and its slope. The general form is: \[ y - y_1 = m(x - x_1) \] Here,

• \((x_1, y_1)\) is a point on the line.
• \(m\) is the known slope of the line.

Once you have the slope, you can plug in the values to form the equation. For the previous example with the points (1,3) and slope -5, choose the point (1, 3) to plug into the formula: \[ y - 3 = -5(x - 1)\] This equation represents the line passing through the point (1, 3) with a slope of -5. Simplifying it gives us a more user-friendly form of the equation.

intercept form

Intercept form, also known as the standard form of a line, is expressed as: \[ Ax + By = C \] This form is very useful when you want to easily determine where the line crosses the x-axis and y-axis. To convert an equation from the point-slope form to the intercept form, you need to rearrange and simplify. For the example we are working on, we have the equation from the last step: \[ y = -5x + 8 \] To convert this to intercept form, follow these steps:

• Move all terms to one side of the equation: \( y + 5x = 8 \).
• Rewrite it to clearly show \( Ax + By = C \): \( 5x + y = 8 \).

In this standard form, it's easy to see that the line crosses the y-axis at 8 and the x-axis can be found easily by setting \(y\) to 0 and solving for \(x\). The intercept form is very practical for many algebraic applications.