Problem 14 - Algebra and Trigonometry

Question

11–18 ? Find the x- and y-intercepts of the graph of the equation. $$ y-2 x y+2 x=1 $$

Step-by-Step Solution

Answer

The x-intercept is \( \left( \frac{1}{2}, 0 \right) \) and the y-intercept is \( (0, 1) \).

Step 1: Simplify the Equation

Start by simplifying the given equation \( y - 2xy + 2x = 1 \). We'll rearrange the terms to make it easier to find intercepts. Combine like terms, if possible, to represent it in a standard linear form. However, due to its structure, we'll keep it as it is for now since it's necessary for solving for intercepts.

Step 2: Finding the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation. Substitute and solve the equation: \[ 0 - 2x(0) + 2x = 1 \] which simplifies to: \[ 2x = 1 \] Divide both sides by 2: \[ x = \frac{1}{2} \] Thus, the x-intercept is \( \left( \frac{1}{2}, 0 \right) \).

Step 3: Finding the y-intercept

To find the y-intercept, set \( x = 0 \) in the original equation. Substitute and solve: \[ y - 2(0)y + 2(0) = 1 \] which simplifies to: \[ y = 1 \] Thus, the y-intercept is \( (0, 1) \).

Key Concepts

x-intercept

An x-intercept is the point where a graph intersects the x-axis on a Cartesian coordinate system. At this point, the value of y is always 0. Understanding x-intercepts is essential, especially when solving linear equations.

To find the x-intercept of an equation, you set the variable y to 0 in the equation, then solve for x.

• This results in the coordinates \((x, 0)\).

In the given example, the original equation is \( y - 2xy + 2x = 1 \).
When we set \( y = 0 \), the equation simplifies to \( 2x = 1 \). By solving this equation for x, we obtain \( x = \frac{1}{2} \).
Hence, the x-intercept is \( \left( \frac{1}{2}, 0 \right) \).
Recognizing x-intercepts can help in graphing equations and understanding how the graph behaves.

y-intercept

The y-intercept is the point where a graph crosses the y-axis in a coordinate system. At this intercept, the value of x is always 0. Finding the y-intercept is equally as important, especially when dealing with linear equations.

To find the y-intercept, you set x to 0 in the equation and then solve for y.

• This will give you the point \((0, y)\).

In our example equation, \( y - 2xy + 2x = 1 \), when we assign \( x = 0 \), it simplifies to \( y = 1 \).
The calculations show that the y-intercept is \((0, 1)\).
Understanding the y-intercept helps in plotting graphs and providing insights into the equation's behavior at the y-axis.

linear equation

Linear equations are a fundamental concept in algebra and can often be represented in the standard form \( Ax + By = C \). They describe a straight line in a two-dimensional space.
Linear equations can have one or two variables, and they represent constant rates of change.

Key characteristics of linear equations include:

• They graph as straight lines on a plane.
• The slope-intercept form \(y = mx + b\) shows the slope (m) and y-intercept (b) directly.
• Detectable x and y-intercepts, which are crucial for graphing the equation.

In the exercise, the equation \( y - 2xy + 2x = 1 \) is a bit unconventional but can still be approached using linear principles for finding intercepts. Simplification or comparison with standard forms helps identify intercepts and interpret the equation.
Understanding linear equations allows students to solve a wide range of algebra problems effectively and accurately.