91Ó°ÊÓ

The x-intercept of an equation is the point where the graph of the equation crosses the x-axis. To find this, we focus on the idea that the y-coordinate at this point is always zero.
For any equation, to find the x-intercept, we simply set \( y = 0 \) and then solve the equation for \( x \).
In the original exercise, we are given the equation \(x^{2} y - x^{2} + 4y = 0\). We substitute \(y = 0\) into the equation like this:

• \(x^{2} (1-y) + 4y = 0\)
• Substituting \( y = 0 \) gives: \( x^{2} (1-0) + 4 \times 0 = x^{2} = 0 \)

This means that the x-intercept is at \(x = 0\), resulting in the x-intercept point (0,0).
Understanding this concept helps in graphing equations. By finding x-intercepts, you identify key points where the graph will touch the horizontal axis.

The y-intercept of an equation is the point where the graph crosses the y-axis. At this point, the x-coordinate will always be zero. Therefore, to find the y-intercept, we substitute \( x = 0 \) in the equation and solve for \( y \).
Looking back at the given equation, \(x^{2} y - x^{2} + 4y = 0\), we proceed by setting \( x = 0 \):

• Substitute \( x = 0 \) to obtain: \( 0 \times y - 0 + 4y = 4y = 0 \)
• Solve for \( y \): \( y = 0 \)

This indicates that the y-intercept is also at the origin (0,0).
By determining the y-intercept, we get another vital point that reflects where the graph meets the vertical axis. Knowing both intercepts greatly simplifies sketching the basic shape of a graph.

Solving algebraic equations involves finding the values of variables that satisfy the equation. This is fundamental in algebra and can be done through various methods depending on the equation's complexity.
In the exercise provided, the equation is \(x^{2} y - x^{2} + 4y = 0\). To solve for intercepts, we used simple substitution and basic algebraic manipulation. Let's briefly recap the process:

• For the x-intercept, set \( y = 0 \), resulting in a simpler equation \( x^2 = 0 \) leading to \( x = 0 \)
• For the y-intercept, set \( x = 0 \), simplifying to \( 4y = 0 \) which gives \( y = 0 \)

These methods highlight key algebraic operations such as substitution and solving for unknowns—essential skills in algebra.
Grasping these techniques is crucial as they lay the groundwork for solving more complex equations encountered in mathematics. Mastery involves practice and understanding which operations simplify an equation gradually.