Problem 14 - Algebra 1

Question

Write an equation of the line that passes through the given points. $$ (-17,-8),(-7,-4) $$

Step-by-Step Solution

Answer

The equation of the line passing through the points (-17,-8) and (-7,-4) is \(y = 0.4x - 1.2\).

Step 1: Calculate the Slope

The slope of a line \(m\) passing through points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m = (y_2 - y_1) / (x_2 - x_1)\). Substituting the given points (-17,-8) and (-7,-4) to calculate the slope: \(m = (-4 - (-8)) / (-7 - (-17)) = 4/10 = 0.4\)

Step 2: Find the Equation of the Line

The equation of the line is given by substituting the calculated slope and one of the given points (\(-17,-8\)) in the line equation \(y = mx + c\). We plug in the values to the definition of the slope, and after simplifying, we find c, the y-intercept. \(-8 = 0.4*-17 + c\), so \(c = -8 + 6.8 = -1.2\)

Step 3: Provide the Final Line Equation

Now that we have the values for the slope \(m\) and the y-intercept \(c\), we can substitute them into the line equation \(y = mx + c\) to obtain the final equation of the line. The equation of the line passing through the points (-17,-8) and (-7,-4) is therefore \(y = 0.4x - 1.2\)

Key Concepts

Slope Calculation

The slope of a line is a measure of how steep the line is, which can also be viewed as the rate at which the line rises or falls as it moves from left to right. In simpler terms, it represents how much the 'y' value (vertical change) changes for a one unit increase in the 'x' value (horizontal change).

To calculate the slope, often denoted as 'm', you take two points on the line, \( (x_1, y_1) \) and \( (x_2, y_2) \), and use the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. For example, given the points \( (-17,-8) \) and \( (-7,-4) \), you would calculate the slope as \[ m = \frac{-4 - (-8)}{-7 - (-17)} = \frac{4}{10} = 0.4 \].
A positive slope indicates an upward trend, while a negative slope indicates a downward trend. In our example, the slope is positive, meaning the line rises as it moves to the right.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. It is an integral part of the line's equation and can be denoted as 'c' in the slope-intercept form of a linear equation, which is \( y = mx + c \). The y-intercept represents the value of 'y' when 'x' is equal to zero.

To find the y-intercept, you can use one of the points through which the line passes and the slope you've already calculated. For instance, using the point \( (-17, -8) \) and the slope \( 0.4 \) from our example, you would plug these into the slope-intercept equation and solve for 'c':
\[\begin{align*}-8 &= 0.4\times-17 + c \Rightarrow \-8 &= -6.8 + c \Rightarrow \-8 + 6.8 &= c \Rightarrow \-1.2 &= c\end{align*}\]Therefore, the y-intercept of our line is \( -1.2 \). This tells us that the line will cross the y-axis at the point (0, -1.2).

Linear Equations

A linear equation represents a straight line on a graph. The most common form of a linear equation is the slope-intercept form, \( y = mx + c \), where 'm' is the slope of the line and 'c' is the y-intercept. Linear equations are used to describe relationships where there is a constant rate of change.

With the slope and y-intercept in hand, writing the equation of our line is straightforward. Our calculated slope is \( 0.4 \) and the y-intercept is \( -1.2 \), which gives us the final equation \( y = 0.4x - 1.2 \). This equation can be used to find the 'y' value for any given 'x' value on the line.
Linear equations are fundamental in various fields, including physics, economics, and engineering, due to their simplicity and the fact that they are easy to manipulate algebraically.