Problem 39 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the points. $$ (14,-3),(-6,9) $$

Step-by-Step Solution

Answer

The equation of the line is \(y = -0.6x + 5.4\)

Step 1: Calculate the Slope

First, calculate the slope (m) of the line using the formula for the slope which is \((y2 - y1) / (x2 - x1)\). Substituting the coordinates of the given points, you get \(m = (9 - (-3)) / (-6 - 14) = 12 / -20 = - 0.6\).

Step 2: Find the y-intercept

Given the slope, we find the y-intercept (b) by substituting the slope and one of the points into the slope-intercept equation and solving for \(b\). Using the point (14,-3), the formula \(y = mx + b\) becomes \(-3 = - 0.6 * 14 + b\), which simplifies to \(b = -3 + 0.6 * 14 = 5.4\).

Step 3: Write the Equation

Substitute the calculated slope and y-intercept into the slope-intercept form equation. The final equation of the line thus is \(y = -0.6x + 5.4\).

Key Concepts

Calculating Slope

Understanding how to calculate the slope is essential for graphing linear equations and analyzing the rate of change between data points. The slope represents the steepness and direction of a line. To find the slope, you use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) where \(x_1, y_1\) and \(x_2, y_2\) are coordinates of two distinct points on the line. A positive slope indicates that the line rises from left to right, while a negative slope means the line falls from left to right. If the slope is zero, the line is horizontal, denoting no change. Conversely, an undefined slope (a division by zero scenario) corresponds to a vertical line.

In the given exercise, you calculated the slope using the points \( (14, -3) \) and \( (-6, 9) \). The formula revealed a slope of \( -0.6 \) after performing the subtraction and division, thus the line falls as it moves from left to right.

Y-intercept

The y-intercept is a fundamental attribute of linear equations, representing the point where the line crosses the y-axis. In the slope-intercept form equation \(y = mx + b\), \(b\) stands for the y-intercept. It provides a starting value for the line when \(x = 0\). To find the y-intercept from two points, one method is to first calculate the slope (as discussed in the previous section), then use one of the given points and substitute the values into the slope-intercept equation to solve for \(b\).

In this exercise, after determining the slope, the equations \(y = mx + b\) was utilized with the point \( (14, -3) \) to find the y-intercept of \( 5.4 \). This means that the line will cross the y-axis at the point \( (0, 5.4) \) on the Cartesian grid.

Linear Equations

Linear equations create the blueprints for lines on a Cartesian coordinate system. They are called 'linear' because they always graph to a straight line. The standard form of a linear equation is \( Ax + By = C \), but for the ease of graphing and understanding its characteristics, it's often converted to the slope-intercept form, \( y = mx + b \). The variable \(m\) denotes the slope, and \(b\) identifies the y-intercept, succinctly encapsulating both the tilt and vertical positioning of the line.

To write the equation in slope-intercept form, just insert the values for the slope and y-intercept which you have calculated. In the example used in the solution, the final equation \( y = -0.6x + 5.4 \) portrays a line with a negative slope, indicating a downward trajectory as one moves along the x-axis, and it crosses the y-axis just above the 5th unit.