Problem 39 - Calculus Single Variable

Question

A general linear equation of a line is given. Find the \(x\) -intercept, the \(y\) -intercept, and the slope of the line. $$ x-3 y=3 $$

Step-by-Step Solution

Answer

The line has an \(x\)-intercept at (3, 0), a \(y\)-intercept at (0, -1), and a slope of \(\frac{1}{3}\).

Step 1: Identify the General Form of the Equation

The equation given is of the form: \[x - 3y = 3\] This equation is in the general form of a linear equation, which can be written as: \[Ax + By = C\] where \(A = 1\), \(B = -3\), and \(C = 3\).

Step 2: Find the x-intercept

To find the \(x\)-intercept, set \(y = 0\) in the equation and solve for \(x\):\[x - 3(0) = 3\] \[x = 3\] Thus, the \(x\)-intercept is \((3, 0)\).

Step 3: Find the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation and solve for \(y\):\[0 - 3y = 3\] \[-3y = 3\] \[y = -1\] Thus, the \(y\)-intercept is \((0, -1)\).

Step 4: Calculate the Slope

For a linear equation in the form \(Ax + By = C\), the slope \(m\) is given by the formula:\[m = -\frac{A}{B}\] Substitute \(A = 1\) and \(B = -3\):\[m = -\frac{1}{-3} = \frac{1}{3}\] Thus, the slope of the line is \(\frac{1}{3}\).

Key Concepts

X-Intercept

The x-intercept of a line is a point where the line crosses the x-axis. This means that at the x-intercept, the y-coordinate is always zero. To find the x-intercept for any equation, you simply set the y-value to zero and solve for x. For our given equation, which is:\[ x - 3y = 3 \], we set \( y = 0 \). This results in the equation \( x - 3(0) = 3 \), which simplifies to \( x = 3 \). Therefore, the x-intercept for this particular line is the point \((3, 0)\). By understanding how to find the x-intercept, you can quickly determine where a line will cross the horizontal axis. This is a crucial intercept for graphing and interpreting linear equations.

Y-Intercept

The y-intercept is similar to the x-intercept, but in this case, it is where the line crosses the y-axis. Here, the x-coordinate is zero. To determine the y-intercept in any linear equation, substitute \( x = 0 \) into the equation and solve for y. Taking our equation:\[ x - 3y = 3 \], setting \( x = 0 \), gives us \( 0 - 3y = 3 \). Solving for y yields \( y = -1 \). Therefore, the y-intercept of this line is \( (0, -1) \). This point is essential when sketching the line on a graph since it tells you where it will meet the vertical axis.

Slope

The slope of a line describes its steepness or incline. In mathematical terms, the slope is the rate at which y changes with respect to x. It is often denoted by \( m \). For a linear equation in the general form \( Ax + By = C \), the slope \( m \) can be calculated by the formula \( m = -\frac{A}{B} \). Now, plugging in our values where \( A = 1 \) and \( B = -3 \), we find the slope to be \( m = -\frac{1}{-3} = \frac{1}{3} \).

• This positive slope indicates the line rises from left to right.
• A slope of zero would mean the line is horizontal.
• If the slope were negative, the line would descend from left to right.

The slope provides insight into how rapidly or slowly a line increases or decreases as you move along the x-axis.