Problem 68 - Algebra for College Students

Question

Find all intercepts for each line. Some of these lines have only one intercept. $$x+2 y=100$$

Step-by-Step Solution

Answer

The intercepts are (100, 0) and (0, 50).

Step 1: Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. Starting with the given equation:\[ x + 2y = 100 \]Set \( y = 0 \):\[ x + 2(0) = 100 \]Simplify to obtain \( x = 100 \)Thus, the x-intercept is at the point (100, 0).

Step 2: Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. Starting again with the given equation:\[ x + 2y = 100 \]Set \( x = 0 \):\[ 0 + 2y = 100 \]Simplify and solve for y:\[ 2y = 100 \]\[ y = 50 \]Therefore, the y-intercept is at the point (0, 50).

Key Concepts

x-intercept

The x-intercept is where a line crosses the x-axis. At this point, the value of y is always zero.
To find the x-intercept of the linear equation \(x + 2y = 100\), we set \(y\) to zero and solve for \(x\).
Starting with the given equation, set \(y = 0\):
\[ x + 2(0) = 100 \]
Simplify to get:
\[ x = 100 \]
This means the x-intercept is at the point (100, 0).
To summarize:

• The x-intercept occurs where \(y = 0\).
• Substitute \(y = 0\) into the equation to find \(x\).
• The x-intercept for this line is at \(x = 100\).

y-intercept

The y-intercept is where a line crosses the y-axis. At this point, the value of x is always zero.
To determine the y-intercept of the linear equation \(x + 2y = 100\), we set \(x\) to zero and solve for \(y\).
Starting with the given equation, set \(x = 0\):
\[ 0 + 2y = 100 \]
Simplify and solve for \(y\):
\[ 2y = 100 \]
\[ y = 50 \]
This means the y-intercept is at the point (0, 50).
Key points to remember:

• The y-intercept occurs where \(x = 0\).
• Substitute \(x = 0\) into the equation to find \(y\).
• The y-intercept for this line is at \(y = 50\).

linear equations

Linear equations describe relationships between two variables with a straight line when graphed. They are typically written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
The general steps to analyze a linear equation such as \(x + 2y = 100\) are:

• Identify the coefficients of \(x\) and \(y\), and the constant term.
• To find intercepts, set one variable to zero and solve for the other.
• Graph the equation using the intercepts to see how the line behaves.

Understanding linear equations is essential as they form the basics of algebra and serve as the foundation for more complex mathematical concepts.
Always remember these key attributes of linear equations:

• They create straight lines on a graph.
• Only have powers of one for both variables \(x\) and \(y\).
• Finding intercepts helps to easily visualize the line's placement on the graph.