Problem 109 - Algebra for College Students

Question

Exploration The intercept form for the equation of a line is 2 \(\frac{x}{a}+\frac{y}{b}=1\) where neither \(a\) nor \(b\) is zero. a) Find the \(x\) - and \(y\) -intercepts for \(\frac{x}{4}+\frac{y}{6}=1\) b) Find the \(x\) - and \(y\) -intercepts for \(\frac{x}{a}+\frac{y}{b}=1\) c) Write the equation of the line through \((0,3)\) and \((-5,0)\) in intercept form. d) Which lines cannot be written in intercept form?

Step-by-Step Solution

Answer

a) x-intercept: 4, y-intercept: 6. b) x-intercept: a, y-intercept: b. c) \ \( \frac{x}{-5} + \frac{y}{3} = 1 \). d) Lines parallel to x-axis or y-axis.

Step 1 - Find the x-intercept for \ \( \frac{x}{4}+\frac{y}{6}=1\)

The x-intercept is found by setting y = 0 in the equation \ \(\frac{x}{4}+\frac{y}{6}=1\).\ When \( y = 0 \): \ \( \frac{x}{4} = 1 \) \ Solving for x, we get: \ \[ x = 4 \]

Step 2 - Find the y-intercept for \ \( \frac{x}{4}+\frac{y}{6}=1\)

The y-intercept is found by setting x = 0 in the equation \ \(\frac{x}{4}+\frac{y}{6}=1\).\ When \( x = 0 \): \ \( \frac{y}{6} = 1 \) \ Solving for y, we get: \ \[ y = 6 \]

Step 3 - Intercepts for \ \( \frac{x}{a} + \frac{y}{b} = 1 \)

The x-intercept is found by setting \( y = 0 \) in the equation \ \( \frac{x}{a} + \frac{y}{b} = 1 \). \ \( \frac{x}{a} = 1 \) \ Solving for x, we get: \ \[ x = a \] \ The y-intercept is found by setting \(x = 0\) in the equation \ \( \frac{x}{a} + \frac{y}{b} = 1 \). \ \( \frac{y}{b} = 1 \) \ Solving for y, we get: \ \[ y = b \]

Step 4 - Write the equation of the line through (0,3) and (-5,0)

Using the points (0, 3) and (-5, 0) as the intercepts, we can write the intercept form: \ \( \frac{x}{-5} + \frac{y}{3} = 1 \)

Step 5 - Identify lines that cannot be written in intercept form

Lines that are parallel to either the x-axis or y-axis cannot be written in intercept form because for these lines either a or b would be infinity, which is not allowed. An example is the horizontal line y = k or the vertical line x = k.

Key Concepts

x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. This point has coordinates \(x, 0\), meaning the y-coordinate is zero. To find the x-intercept in any line equation, you need to set y to 0 and solve for x. For example, when given the equation \(\frac{x}{4} + \frac{y}{6} = 1\), set \(y = 0\):

\ \frac{x}{4} = 1 \
Solving this, you get:

\ x = 4 \
So, the x-intercept is (4, 0). This method works similarly for all linear equations using the intercept form.

y-intercept

The y-intercept is where the line crosses the y-axis, meaning it occurs at the point (0, y). To find the y-intercept, set x equal to 0 in the line's equation and solve for y. For the equation \(\frac{x}{4} + \frac{y}{6} = 1\), set \(x = 0\):

\ \frac{y}{6} = 1 \
Solving this, you get:

\ y = 6 \
So, the y-intercept is (6, 0). The same method is applied to other lines in intercept form.

parallel lines

Parallel lines are lines in a plane that never meet. They have the same slope but different y-intercepts. For lines given in intercept form, they cannot be parallel unless parameters are set accordingly. Parallel lines to the x-axis are horizontal (e.g., y = k) and have no x-intercept. Similarly, lines parallel to the y-axis are vertical (e.g., x = k) and have no y-intercept. These lines cannot be written in intercept form as their intercept values would either be undefined or infinite, which is not possible in the intercept form equation.

line equation

The line equation tells you about the relationship between the x and y coordinates on a straight line. The intercept form of a line is given as \(\frac{x}{a} + \frac{y}{b} = 1\), where 'a' is the x-intercept and 'b' is the y-intercept. To understand this better, consider the equation of a line passing through two intercept points, such as (0,3) and (-5,0). The intercept form equation would be:

\ \frac{x}{-5} + \frac{y}{3} = 1 \
This equation tells us that the line cuts the x-axis at -5 and the y-axis at 3. Intercept form helps to define the points where the line crosses the axes, making graphing easier.