Problem 102 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write an equation of the line passing through the two points. Use the slope- intercept form, if possible. If not possible, explain why. $$\left(\frac{3}{5}, 0\right),(4,6)$$

Step-by-Step Solution

Answer

The equation of the line passing through the points \((3/5, 0)\) and \((4,6)\) is \(y = 3x - 6\).

Step 1: Calculating Slope

The slope (\(m\)) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). Substitute the given points into the formula: \(m = (6 - 0) / (4 - 3/5) = 6 / 2 = 3\). Therefore, the slope is 3.

Step 2: Calculating y-Intercept

We can use the slope we just found, along with one of the points from the problem to solve for \(b\) in the equation \(y = mx + b\). Substituting \(m = 3\) and the point (4, 6), we get \(6 = 3*4 + b\), so \(b = 6-12 = -6\). Therefore, the y-intercept is -6.

Step 3: Writing the Equation

The slope-intercept form of the equation is \(y = mx + b\). Substituting \(m = 3\) and \(b = -6\), we get \(y = 3x - 6\). This is the equation of the line that passes through the given points.

Key Concepts

Calculating Slope

Understanding how to calculate the slope is crucial for analyzing the rate at which one variable changes with respect to another in a linear relationship. The slope represents the steepness of a line and is often referred to by the letter 'm'. To find it, you need two points on the line, expressed as \( (x_1, y_1) \) and \( (x_2, y_2) \) .

The formula to calculate slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). By substituting the coordinates of our two points into this formula, we can determine the line's slope. For example, if our points are \( \left(\frac{3}{5}, 0\right) \) and \( (4,6) \) , we plug them into the formula like this: \( m = \frac{6 - 0}{4 - \frac{3}{5}} = \frac{6}{\frac{17}{5}} = 3 \), so the slope is 3. This tells us that for every one-unit increase in 'x', the 'y' value increases by three units.

Finding Y-intercept

The y-intercept is the point where the line crosses the y-axis. It is usually represented by the variable 'b'. Once you know the slope of the line, you can use any point on the line to find the y-intercept with the formula \( y = mx + b \).

Let's take the point \( (4,6) \) and the slope we've already calculated, 3. We insert these into the slope-intercept equation to solve for 'b': \( 6 = 3\times 4 + b \), leading us to \( b = 6 - 12 = -6 \). There we have it, the y-intercept is -6, meaning our line crosses the y-axis at \( (0, -6) \) .

Finding the y-intercept is fundamental because it gives us a starting point to draw the entire line on a graph or to understand where a linear relationship begins.

Writing Linear Equations

Writing linear equations is a way of expressing the relationship between two variables in a straight line on a coordinate plane. The most common form used is the slope-intercept form, \( y = mx + b \) , where 'm' is the slope and 'b' is the y-intercept. This equation tells you how to find the y-value for any x-value on the line.

With the slope (3) and the y-intercept (-6) from our previous steps, we write the equation of our line as \( y = 3x - 6 \). This equation is simple to use and visually represents the linear relationship on a graph. Knowing how to write linear equations empowers students to model and solve real-world problems involving linear relationships.

Slope of a Line

The concept of slope is integral to understanding the direction and steepness of a line on a graph. As we discussed in the first section, slope is calculated as the 'rise over run' between two points. It provides a numerical value to the inclination angle of the line.

A positive slope means that the line rises as it moves from left to right, indicating a direct relationship between 'x' and 'y'. Conversely, a negative slope means the line falls as it moves from left to right, indicating an inverse relationship. If the slope is zero, the line is horizontal, revealing that 'y' does not change regardless of 'x'. Understanding the slope of a line allows students to predict and make assumptions about the data presented in a linear graph.