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Chapter 0: Problem 19

Finding Intercepts In Exercises \(19-28,\) find any intercepts. \(y=2 x-5\)

Short Answer

Expert verified
The x-intercept is (2.5,0) and the y-intercept is (0,-5).

Step by step solution

01

Calculating the x-intercept

To find the x-intercept, in the given line equation \(y = 2x - 5\), set \(y = 0\) and solve for x. This will give you the x-coordinate of the x-intercept.
02

Solve for x

Plugging y = 0 into the equation gives \(0 = 2x - 5\). Solving this linear equation for x gives \(x = 2.5\). So the x-intercept is (2.5,0).
03

Calculating the y-intercept

To find the y-intercept, in the given line equation \(y = 2x - 5\), set \(x = 0\) and solve for y. This will give you the y-coordinate of the y-intercept.
04

Solve for y

Plugging x = 0 into the equation gives \(y = -5\). So the y-intercept is (0,-5).

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Key Concepts

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

Understanding the x-intercept
Finding the x-intercept of a line is all about identifying where the graph of the equation crosses the x-axis.
When a line crosses the x-axis, the y-value is zero because the point lies on the horizontal axis, where the vertical distance is zero.
To find the x-intercept, you set the y variable to 0 in the given equation and solve for x.
  • For the line equation given, which is \( y = 2x - 5 \), setting \( y = 0 \) transforms the equation into \( 0 = 2x - 5 \).
  • Next, solve the equation for x by isolating x on one side. This results in \( 2x = 5 \), and dividing both sides by 2 gives \( x = 2.5 \).
The x-intercept is then shown as a point on the coordinate plane, which is \( (2.5, 0) \), indicating that the line crosses the x-axis at 2.5.
Understanding the y-intercept
Finding the y-intercept involves discovering where the graph crosses the y-axis.
This point is significant because it shows where the line meets the vertical axis at a horizontal distance of zero; hence the line crosses the y-axis.
For our equation \( y = 2x - 5 \), setting \( x = 0 \) helps find this specific point.
  • With \( x = 0 \), the equation simplifies to \( y = -5 \).
Therefore, the y-intercept of the line is \((0, -5)\), which means the line intersects the y-axis at -5.
In visual terms, this is the starting point of the line if imagined as a slope starting from the vertical axis.
This process helps in sketching the line on a graph by providing a tangible starting point.
Solving linear equations for intercepts
Linear equations are equations that create straight lines when graphed.
Understanding how to solve these for intercepts is vital because it provides a clear picture of the line's path on a coordinate plane.
Intercepts, namely the x-intercept and y-intercept, offer specific points where the line crosses the axes.
  • The x-intercept is found by setting the equation's y to 0, simplifying the task to calculating where the path of the line meets the x-axis.
    This involves straightforward algebraic manipulation to determine the x-coordinate where the y-value becomes zero.
  • Similarly, the y-intercept is discovered by setting x to 0, and it denotes where the line crosses the y-axis.
    This is calculated by simplifying the equation to find the y-coordinate when the x-value is zero.
These intercepts are crucial for graphing because they provide exact crossing points for the line, which are essential for understanding and predicting the line's trajectory and behavior.

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Most popular questions from this chapter

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