Problem 69 - 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. $$ \frac{x}{5}+\frac{y}{7}=1 $$

Step-by-Step Solution

Answer

The x-intercept is 5 and the y-intercept is 7. Constants \(a\) and \(b\) represent the x- and y-intercepts, respectively.

Step 1: Understand the intercept form of a line

The intercept form of a line is given by \( \frac{x}{a} + \frac{y}{b} = 1 \). This equation indicates a line where \( a \) and \( b \) represent the intercepts on the \( x\)-axis and the \( y\)-axis, respectively.

Step 2: Identify the constants a and b

In the given problem, the equation is \( \frac{x}{5} + \frac{y}{7} = 1 \). Here, \( a = 5 \) and \( b = 7 \). Thus, \( a \) and \( b \) are the intercepts.

Step 3: Calculate the x-intercept

To find the \( x\)-intercept, set \( y = 0 \) in the equation and solve for \( x \). Substitute \( y = 0 \) into \( \frac{x}{5} + \frac{0}{7} = 1 \) which simplifies to \( \frac{x}{5} = 1 \). Solving for \( x \), we have \( x = 5 \). Thus, the \( x\)-intercept is 5.

Step 4: Calculate the y-intercept

To find the \( y\)-intercept, set \( x = 0 \) in the equation and solve for \( y \). Substitute \( x = 0 \) into \( \frac{0}{5} + \frac{y}{7} = 1 \) which simplifies to \( \frac{y}{7} = 1 \). Solving for \( y \), we have \( y = 7 \). Thus, the \( y\)-intercept is 7.

Step 5: Draw conclusions about what a and b represent

From the calculations in Steps 3 and 4, we found that the constants \( a = 5 \) and \( b = 7 \) represent the \( x\)-intercept and \( y\)-intercept, respectively, of the line. This is consistent with the definition of the intercept form of a line.

Key Concepts

Understanding X-Intercept

The x-intercept is the point where a line touches the x-axis. This is where the y-value is zero. For the equation in intercept form, simplify the process: just set the y-value to zero. This simplifies the equation a lot. For example, consider the intercept form equation \(\frac{x}{5} + \frac{y}{7} = 1\). To find the x-intercept:

• Set \(y = 0\)
• The equation becomes \(\frac{x}{5} = 1\)
• Solving gives \(x = 5\)

Thus, the x-intercept is 5.This shows that when y is zero, x equals a constant number \(a\), which in this case is 5. This helps us understand that this constant represents the specific point where the line meets the x-axis.

Exploring Y-Intercept

Similar to the x-intercept, the y-intercept is found where the line crosses the y-axis. This means the x-value is zero. Using the intercept form \(\frac{x}{5} + \frac{y}{7} = 1\), find the y-intercept by:

• Setting \(x = 0\)
• The equation simplifies to \(\frac{y}{7} = 1\)
• Solving gives \(y = 7\)

Hence, the y-intercept is 7.It becomes clear that the constant b (which is 7 in this situation) indicates where the line intersects the y-axis. This point illustrates the y-value when the line meets the axis, giving a meaningful representation of the line's slope in relation to the y-axis.

Grasping Linear Equations in Intercept Form

Linear equations are fundamental in algebra and show how two variables relate in a straight line within a graph. The beauty of the intercept form is its straightforward representation. The equation \(\frac{x}{a} + \frac{y}{b} = 1\) conveniently displays both intercepts directly:

• \(a\) represents the x-intercept
• \(b\) represents the y-intercept

This makes it easy to identify how a line behaves on a graph. The intercept form is excellent for quickly drawing a line since you immediately know where it crosses both axes. Recognizing what \(a\) and \(b\) mean provides instant insight into the nature of the line, thus simplifying graphing and solving tasks for students who engage with these concepts.