Problem 72 - College Algebra with Modeling and Visualization

Question

The intercept form of a line is \(\frac{x}{a}+\frac{y}{b}=1\) Determine the \(x\) - and \(y\) -intercepts on the graph of the equation. Draw a conclusion about what the constants a and b represent in this form. $$ \frac{5 x}{6}-\frac{y}{2}=1 $$

Step-by-Step Solution

Answer

The x-intercept is \(\frac{6}{5}\), and the y-intercept is 2; thus, \(a\) is the x-intercept and \(b\) is the y-intercept.

Step 1: Identify the Given Equation

The given equation is: \( \frac{5x}{6} - \frac{y}{2} = 1 \). This equation is not in the standard intercept form \( \frac{x}{a} + \frac{y}{b} = 1 \). We need to rewrite it in this form.

Step 2: Rewrite the Equation in Intercept Form

We want the equation as \( \frac{x}{a} + \frac{y}{b} = 1 \).First, rearrange the terms: \( \frac{5x}{6} = 1 + \frac{y}{2} \).To simplify terms, multiply through by 6 to clear the fractions: \[ 5x = 6 + 3y. \]Next, reorganize it as: \[ 5x - 3y = 6. \]Divide every term by 6: \[ \frac{5x}{6} - \frac{3y}{6} = 1. \]Simplify the second term: \( \frac{5x}{6} + \frac{y}{2} = 1 \).

Step 3: Determine the x-intercept

The x-intercept is found by setting \(y = 0\) in the intercept form equation: \( \frac{5x}{6} + \frac{0}{2} = 1 \).Solving for \(x\): \( \frac{5x}{6} = 1 \).Multiply both sides by 6 to eliminate the fraction: \( 5x = 6 \).Solve for \(x\): \( x = \frac{6}{5} \).

Step 4: Determine the y-intercept

The y-intercept is found by setting \(x = 0\) in the intercept form equation: \( \frac{0}{6} + \frac{y}{2} = 1 \).Solving for \(y\): \( \frac{y}{2} = 1 \).Multiply both sides by 2 to eliminate the fraction: \( y = 2 \).

Step 5: Interpret the Constants a and b

In the equation \( \frac{x}{a} + \frac{y}{b} = 1 \), the constant \(a\) represents the x-intercept and \(b\) represents the y-intercept.In this case, after rewriting it correctly, we have \( a = \frac{6}{5} \) and \( b = 2 \). The values of \(a\) and \(b\) give us the points where the line intersects the x-axis and y-axis respectively.

Key Concepts

Understanding the x-intercept

The x-intercept of a line on a graph is the point where the line crosses the x-axis. This occurs when the value of y is zero. To find the x-intercept of the equation, we set the variable y to zero and solve for x.
In this specific exercise, when we set y to zero, the intercept form equation becomes \( \frac{5x}{6} + \frac{0}{2} = 1 \). Solving this results in \( 5x = 6 \), hence \( x = \frac{6}{5} \).
It's helpful to understand this concept because the x-intercept provides key insight into how a line behaves on a graph.

• This intercept provides a basic understanding of where the line will cross the x-axis.
• It is directly related to the constant value 'a' in the intercept form, which represents this x-intercept.

Deciphering the y-intercept

The y-intercept is the point where a line crosses the y-axis, occurring when x equals zero. To find it, substitute x with zero in the line’s equation, then solve for y. In our example, the equation \( \frac{0}{6} + \frac{y}{2} = 1 \) simplifies to \( \frac{y}{2} = 1 \).
By solving for y, we find \( y = 2 \), meaning the line crosses the y-axis at this point.
Understanding the y-intercept is crucial because it provides a point through which the line passes on the y-axis. This intercept is associated with the constant 'b' in the equation.

• The y-intercept shows where the graph intersects the y-axis, providing another reference for drawing the line.
• The constant 'b' determines this exact intercept on the y-axis in the intercept equation form.

Algebraic equations and intercept form

Algebraic equations in intercept form allow us to easily deduce the intercepts of a line. The standard form \( \frac{x}{a} + \frac{y}{b} = 1 \) visually conveys how the line interacts with the axes. It breaks down into the separate intercepts and demonstrates their position on the graph.
Key elements of algebraic equations in intercept form include:

• 'a' signifies the x-intercept, depicting where the line meets the x-axis.
• 'b' denotes the y-intercept, illustrating where the line crosses the y-axis.
• This form simplifies understanding of graph movements by showing precise intercept locations.

Using intercept form streamlines graphing and comprehension of how lines pivot around intercepts, making it an invaluable tool in algebra.