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Chapter 1: Problem 12

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ 4 x-2 y-5=0 $$

Short Answer

Expert verified
The x-intercept is \(5/4\) and the y-intercept is \(5/2\).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \(y = 0\) in the given equation and solve for \(x\). Substituting \(y = 0\) in the equation, we have \(4x - 2*0 - 5 = 0\), which simplifies to \(4x - 5 = 0\). Solving this for \(x\) we have \(x = 5/4\). Therefore, the x-intercept is \(5/4\).
02

Find the y-intercept

To find the y-intercept, set \(x = 0\) in the given equation and solve for \(y\). Substituting \(x = 0\) in the equation we have \(4*0 - 2y - 5 = 0\), which simplifies to \(-2y - 5 = 0\). Solving this for \(y\) we get \(y = -5/-2 = 5/2\). Therefore, the y-intercept is \(5/2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

The x-intercept
When we talk about the intercepts of a graph, the x-intercept is the point where the graph of an equation crosses the x-axis. This means that at this point, the value of y is zero. It鈥檚 a crucial concept because it gives us information about where a line hits the horizontal axis, which can tell us about its behavior and position.
To find the x-intercept from the equation, we substitute y = 0 into the equation and solve for x.
  • Start from the equation: \(4x - 2y - 5 = 0\)
  • Set \(y = 0\): \(4x - 2(0) - 5 = 0\)
  • This simplifies to \(4x - 5 = 0\)
  • Solve for \(x\) to find \(x = \frac{5}{4}\)
That鈥檚 it! The x-intercept for this equation is \(\left( \frac{5}{4}, 0 \right)\), a precise spot on the horizontal axis. Understanding this helps us know where the graph starts its journey across the x-axis.
The y-intercept
The y-intercept of a graph is the point where it crosses the y-axis, which means this is the spot where x equals zero. Like the x-intercept, the y-intercept gives us valuable insights into the position of the line, showing us where it strikes the vertical axis.
To find the y-intercept, you set x = 0 and solve for y in the given equation. Here's a step-by-step look at how it's done:
  • Begin with the equation: \(4x - 2y - 5 = 0\)
  • Substitute \(x = 0\): \(4(0) - 2y - 5 = 0\)
  • This simplifies to \(-2y - 5 = 0\)
  • Solve for \(y\), which will be \(y = \frac{5}{2}\)
Once calculated, the y-intercept is found at \(\left(0, \frac{5}{2}\right)\). Knowing both intercepts helps in sketching the graph and understanding its path on a coordinate plane.
Linear equations
Linear equations like \(4x - 2y - 5 = 0\) are equations of a straight line. These equations take on a simple form of \(Ax + By + C = 0\), where A, B, and C are constants. The magic of linear equations lies in their straightforwardness.
A linear equation describes a relationship between x and y that, when plotted, forms a straight line on a graph. This is why they are often a first love in algebra鈥攖hey're predictable and intuitive.
  • The slope of the line, determined as \(-\frac{A}{B}\), depicts the angle and steepness.
  • Intercepts can be found using substitution techniques (like setting y = 0 for x-intercepts).
These properties make linear equations extremely useful in both theoretical and practical situations. Whether estimating trends or solving real-life problems, understanding how to work with a linear equation and its intercepts is integral.

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