Problem 76 - Algebra for College Students

Question

Find all intercepts for each line. Some of these lines have only one intercept. $$2-10 y=0$$

Step-by-Step Solution

Answer

The y-intercept is (0, \( \frac{1}{5}\))There is no x-intercept.

Step 1: Rewrite the Equation in Standard Form

The given equation is: \[2 - 10y = 0\]First, rewrite the equation to isolate the variable term: \[ -10y = -2\]Then, divide both sides by -10:\[ y = \frac{2}{10}\]Simplify:\[ y = \frac{1}{5}\]

Step 2: Identify the Intercept

Since the equation does not involve the variable \(x\), there is no x-intercept. The y-intercept is the value of \(y\) when \(x\) is 0. Here, the y-intercept is \(\frac{1}{5}\), so the only intercept is at \[ (0, \frac{1}{5})\]

Key Concepts

y-intercept

In a linear equation, the y-intercept is the point where the line crosses the y-axis. This occurs when the value of the x variable is zero. Imagine you have a graph, and you draw a vertical line along the y-axis. Where your given line touches this vertical line is the y-intercept.
For instance, in the equation given, \[2 - 10y = 0\], we observe that the equation only involves the variable y.

• This means there will be no x-intercept in the graph.
• By isolating the term with y, we find the y-intercept is \[ (0, \frac{1}{5}) \].

linear equations

Linear equations are equations that form a straight line when graphed on a coordinate plane. They typically include terms that are either constant numbers or products of a constant and a single variable.
A general form of a linear equation in two variables is \[Ax + By + C = 0\].
In this format, A, B, and C represent constants, while x and y are variables that can change.
A few important characteristics of linear equations include:

• They produce straight lines when plotted on a graph.
• The highest power of variables is 1.
• The solutions of these equations make up the points on the line.

solving equations

Solving equations involves finding the value of the variables that make the equation true.
We start with rewriting the equation to simplify and find these values systematically.
For example, given the linear equation \[2 - 10y = 0\], we follow these steps:
1. **Rearrange the equation**: Isolate the variable term, here y, by moving all constants to the other side. This gives us \[-10y = -2\].
2. **Solve for the variable**: Divide both sides by the coefficient of y (which is -10) to find \[y = \frac{-2}{-10}\] which simplifies to \[y = \frac{1}{5}\].
By systematically simplifying the equation, we find that \[y = \frac{1}{5}\], identifying \[ (0, \frac{1}{5}) \] as the intercept.