Problem 71 - 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. $$ \frac{2 x}{3}+\frac{4 y}{5}=1 $$

Step-by-Step Solution

Answer

The x-intercept is \((\frac{3}{2}, 0)\) and the y-intercept is \((0, \frac{5}{4})\). Constants \(a\) and \(b\) are the x- and y-intercepts respectively.

Step 1: Identify Intercept Form

The line's equation given is \(\frac{2x}{3} + \frac{4y}{5} = 1\). This matches the general intercept form of a line \(\frac{x}{a} + \frac{y}{b} = 1\). Here, \(a\) and \(b\) are the coefficients of \(x\) and \(y\) when expressed as fractions with the denominator as the constant 1 on the right-hand side.

Step 2: Determine x-intercept

To find the \(x\)-intercept, set \(y = 0\) in the equation \(\frac{2x}{3} + \frac{4y}{5} = 1\). Substituting, we get \(\frac{2x}{3} = 1\). Solving for \(x\) gives: \[ x = \frac{3}{2} \] Thus, the \(x\)-intercept is \((\frac{3}{2}, 0)\).

Step 3: Determine y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation \(\frac{2x}{3} + \frac{4y}{5} = 1\). Substituting, we get \(\frac{4y}{5} = 1\). Solving for \(y\) gives: \[ y = \frac{5}{4} \] Thus, the \(y\)-intercept is \((0, \frac{5}{4})\).

Step 4: Conclusion about a and b

In the intercept form \(\frac{x}{a} + \frac{y}{b} = 1\), \(a\) represents the \(x\)-intercept and \(b\) represents the \(y\)-intercept of the line. From the given line equation, the coefficients \(\frac{3}{2}\) and \(\frac{5}{4}\) show that \(a = \frac{3}{2}\) and \(b = \frac{5}{4}\), which are indeed the intercepts.

Key Concepts

Understanding X-Intercept

The x-intercept of a line is a crucial concept when analyzing linear equations. To find the x-intercept, you look for the point where the graph of the equation crosses the x-axis. At this point, the y-coordinate will always be zero because it's right on the axis itself.
For example, if you are given the equation \( \frac{2x}{3} + \frac{4y}{5} = 1 \), you can find the x-intercept by setting \( y = 0 \).

• Insert the value into the equation: \( \frac{2x}{3} + \frac{4(0)}{5} = 1 \).


• This simplifies to \( \frac{2x}{3} = 1 \).


• Solving for \( x \), multiply both sides by 3 to get \( 2x = 3 \), and then divide by 2 to find \( x = \frac{3}{2} \).

Therefore, the x-intercept is the point \( (\frac{3}{2}, 0) \). This information helps identify where the line touches the x-axis and is essential for graphing linear equations.

Identifying Y-Intercept

The y-intercept is another key feature when looking at linear equations. It indicates where the graph of the equation intersects the y-axis. At this point, the x-coordinate will always be zero.
To find the y-intercept in the given equation \( \frac{2x}{3} + \frac{4y}{5} = 1 \), set \( x = 0 \).

• Substitute the value into the equation: \( \frac{2(0)}{3} + \frac{4y}{5} = 1 \).


• This simplifies to \( \frac{4y}{5} = 1 \).


• Solving for \( y \), multiply both sides by 5 to get \( 4y = 5 \), and then divide by 4 to obtain \( y = \frac{5}{4} \).

Thus, the y-intercept is \( (0, \frac{5}{4}) \). Knowing this point is crucial for understanding how the line behaves in relation to the y-axis, further illustrating the relationship between x and y.

Exploring Linear Equations

Linear equations are fundamental in mathematics, representing straight lines when graphed. The intercept form of a linear equation, \( \frac{x}{a} + \frac{y}{b} = 1 \), offers a clear way to find both intercepts immediately.
Here are some key features of linear equations in intercept form:

• Simplicity: The equation is straightforward because it directly reveals both the x-intercept \( a \) and y-intercept \( b \). Each intercept can be found by setting the other variable to zero, simplifying the computational process.


• Purpose: Using this form helps to quickly determine where a line crosses the axes, which is essential for graphing and interpreting data in various practical applications.


• Visual Understanding: By knowing the intercepts, one gains insights into the slope and direction of the line without complex calculations.

The form facilitates a deeper intuition for how changes in intercepts affect the graph, thereby aiding in more efficient problem solving.