Problem 32 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (-3,6) and (3,-2)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \( y - 6 = - \frac{4}{3}(x + 3)\) and in slope-intercept form is \(y = -\frac{4}{3}x + 2\).

Step 1: Calculate the Slope

The first step is to calculate the slope of the line passing through the points (-3,6) and (3,-2). The formula used to calculate slope (m) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where (x1, y1) and (x2, y2) are the coordinates of the two given points. Plugging in the given values gets us \( m = \frac{-2-6}{3-(-3)} = - \frac{8}{6} = - \frac{4}{3}\), so the slope of the line is -4/3.

Step 2: Construct Point-Slope Form

Having found the slope and having given two points, the point-slope form of the line can be written using the formula \(y - y_1 = m(x - x_1)\), using (-3,6) as (x1, y1) we get: \( y - 6 = - \frac{4}{3}(x + 3)\).

Step 3: Convert to Slope-Intercept Form

The final step is to convert the point-slope form of the equation into slope-intercept form, \(y = mx + b\), where b is the y-intercept. Simplifying \( y - 6 = - \frac{4}{3}(x + 3)\) results in \(y = -\frac{4}{3}x + 2 \)

Key Concepts

Slope-Intercept Form

When studying linear equations, the slope-intercept form is one of the easiest and most commonly used representations. It is expressed as
\( y = mx + b \),
where \( m \) represents the slope of the line, and \( b \) stands for the y-intercept, which is the point where the line crosses the y-axis. This form is particularly useful because it provides immediate visual information about the slope and y-intercept of a line simply by looking at the coefficients in the equation.

For the example at hand, the slope-intercept form of the line passing through the points (-3,6) and (3,-2) is derived by isolating \( y \) in the point-slope form equation. The simplification process included distributing the slope and adding 6 to both sides to solve for \( y \), resulting in \( y = -\frac{4}{3}x + 2 \), where \( -\frac{4}{3} \) is the slope (\( m \)), and \( 2 \) is the y-intercept (\( b \)). This form quickly tells us that for each unit we move horizontally, the value of \( y \) decreases by \( \frac{4}{3} \), and the line crosses the y-axis at (0, 2).

Calculate Slope

Understanding how to calculate the slope is crucial when dealing with linear equations. The slope determines the steepness and the direction of a straight line.

To find the slope (\( m \)) between two points, the rise over run method is used, which is the change in y-coordinates divided by the change in x-coordinates, following the formula
\( m = \frac{y_2 - y_1}{x_2 - x_1} \).

In the given exercise, we applied this formula to the points (-3,6) and (3,-2), resulting in a slope of \( -\frac{4}{3} \). This negative slope indicates that the line is falling or decreasing from left to right. Furthermore, the magnitude of the slope, \( \frac{4}{3} \), suggests a moderate steepness. Recognizing these properties of the slope helps in graphing the linear equation and understanding its behavior in a graphical context.

Linear Equations

Linear equations form the basis of algebra and are equations of straight lines on a graph. They typically look like \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. A linear equation can be manipulated into different forms such as point-slope, slope-intercept, or standard form—each providing particular insights or solving specific problems.

Point-slope form, once derived using a known point and the slope, is a stepping stone towards writing the equation in slope-intercept form, which is often more intuitive for graphing and analysis purposes. The ability to transition between these forms allows for flexibility in solving and applying linear equations to various problems. The exercise improvement advice focuses on ensuring that students can see both the algebraic manipulation and the geometrical interpretation of these transformations, which is essential for a deep understanding of linear equations.