Problem 31 - Algebra 1

Question

Write an equation of the line that has the given \(x\) -intercept and slope. $$x \text{-intercept} =4, m=3$$

Step-by-Step Solution

Answer

The equation of the line is \(y = 3x - 12\).

Step 1: Understand the given intercept and slope

From the exercise, you know that the slope \(m=3\) and the x-intercept is 4. This means the point of intersection on the x-axis is \((4,0)\).

Step 2: Determine the y-intercept using the slope and a point on the line

The equation of a line given a point \((x_1, y_1)\) and slope \(m\) is given by \(y - y_1 = m(x - x_1)\). From the given x-intercept, we know a point on the line is \((4,0)\). Substitute \(m=3\), \(x_1=4\), and \(y_1=0\) into the line equation to get \(b\), the y-intercept.

Step 3: Write the equation of the line

Upon substituting, you get the y-intercept \(b = -12\). Therefore, the equation of the line in slope-intercept form will be \(y = 3x - 12\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial for graphing and analyzing lines. Simply put, the slope-intercept form is written as:
\(y = mx + b\)
where \(m\) represents the slope of the line, and \(b\) represents the y-intercept, which is where the line crosses the y-axis.

This form is extremely convenient because it gives you the rate of change of the line, which is the slope, and the starting point of the line along the y-axis directly in the equation. In our exercise, the slope given is 3 and, through calculation, we find the y-intercept to be -12. Thus, writing our linear equation in slope-intercept form yields:
\(y = 3x - 12\).

The beauty of this format is its directness and ease of use, especially when graphing by hand or interpreting the line's behavior. For instance, the positive slope indicates an upward trend, meaning as x increases, y increases at a rate of 3 units up for every 1 unit across.

X-Intercept

The x-intercept of a line refers to the point where the line crosses the x-axis. To find the x-intercept, you set the value of \(y\) in the equation of the line to zero and solve for \(x\).

For example, if we take the previously determined slope-intercept form \(y = 3x - 12\) and want to calculate the x-intercept, we set \(y\) to zero:
\(0 = 3x - 12\).
Solving for \(x\), we get \(x = 4\), which matches the given intercept in the exercise. This means our line crosses the x-axis at the point \((4,0)\).

Finding the x-intercept is essential as it provides one crucial point on the line, which helps in plotting the line accurately on a graph. Moreover, x-intercepts can have direct interpretations in various applications, such as break-even points in economics or roots in algebra.

Slope of a Line

The slope of a line, often denoted as \(m\), measures the steepness, incline, or grade of the line. Mathematically, it is the ratio of the change in the y-values to the change in the x-values between two distinct points on the line.

To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), you use the formula:
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
The slope can be positive, negative, zero, or undefined. If the slope is positive, as in the exercise with a slope of 3, the line rises from left to right. Conversely, a negative slope means the line falls from left to right. A zero slope indicates a horizontal line, whereas an undefined slope corresponds to a vertical line.

The concept of slope is foundational in understanding the relationship between variables in an equation. In the context of the step-by-step solution, once we knew the slope (3) and a point on the line (the x-intercept), we could easily determine the full equation of the line using the slope-intercept form.