Problem 74 - Calculus Volume 1

Question

Write the equation of the line satisfying the given conditions in slope- intercept form. \(x\) -intercept \(=-6\) and \(y\) -intercept \(=9\)

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{3}{2}x + 9 \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) represents the y-intercept.

Step 2: Identify Given Intercepts

We are given that the x-intercept is \(-6\) and the \( y \)-intercept is \(9\). The x-intercept means the line crosses the x-axis at the point \((-6, 0)\), and the y-intercept means the line crosses the y-axis at the point \((0, 9)\).

Step 3: Calculate the Slope

The slope \(m\) of a line can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the points \((-6, 0)\) and \((0, 9)\) from the intercepts, we find the slope: \[ m = \frac{9 - 0}{0 + 6} = \frac{9}{6} = \frac{3}{2} \].

Step 4: Substitute into Slope-Intercept Form

Now substitute the slope \( m = \frac{3}{2} \) and the y-intercept \( b = 9 \) into the slope-intercept form equation: \( y = mx + b \). This gives us: \[ y = \frac{3}{2}x + 9 \].

Key Concepts

Linear Equations

Linear equations form the backbone of algebraic study. They describe lines on a graph and are usually expressed in the form of \(y = mx + b\). In this representation, \(m\) is the slope and \(b\) is the y-intercept. This is known as the slope-intercept form. A linear equation can be understood as a balance between variables and constants, creating a straight line when graphed. The simplicity of linear equations makes them a powerful tool in mathematics and real-world applications. They can model anything from the increase of savings over time to the trajectory of an object in motion. Understanding linear equations is essential for solving problems across various disciplines in mathematics and beyond.

X-intercept

The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of \(y\) is zero. To find the x-intercept from a linear equation, you set \(y = 0\) and solve for \(x\). For example, in the problem we are examining, the x-intercept is provided as \(-6\). This means the line crosses the x-axis at the point \((-6, 0)\). Knowing the x-intercept helps in drawing and understanding the line's path on a graph, as it pinpoints a definite location where the line "touches" the horizontal axis.

Y-intercept

The y-intercept is the point where the line crosses the y-axis. Here, the value of \(x\) is zero. Identifying the y-intercept is straightforward when a line is presented in slope-intercept form \(y = mx + b\), as it is simply the constant \(b\). In our example, the y-intercept is given as \(9\), so the line crosses the y-axis at point \((0, 9)\). This intersection plays a significant role because it defines one of the two crucial reference points used to graph a line and is necessary for understanding the behavior of the function represented by the line.

Slope Calculation

The slope of a line represents its steepness or the rate of change. It is calculated from two points on the line using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), which measures the change in \(y\) over the change in \(x\). In our specific problem, we use the points \((-6, 0)\) and \((0, 9)\) to determine the slope. Substituting these into the formula provides the slope \( m = \frac{9 - 0}{0 + 6} = \frac{3}{2} \). The slope tells us that for every 2 units moved horizontally, the line moves 3 units vertically. Understanding slope is key to interpreting the direction and steepness of a line on a graph.