Problem 40 - Calculus Single Variable

Question

A general linear equation of a line is given. Find the \(x\) -intercept, the \(y\) -intercept, and the slope of the line. $$ 5 x-2 y=10 $$

Step-by-Step Solution

Answer

The x-intercept is (2,0), the y-intercept is (0,-5), and the slope is \(\frac{5}{2}\).

Step 1: Identify the Standard Form of the Line Equation

The given equation is \(5x - 2y = 10\), which is in the standard form \(Ax + By = C\). Here, \(A = 5\), \(B = -2\), and \(C = 10\).

Step 2: Find the X-Intercept

The x-intercept occurs when \(y = 0\). Substitute \(y = 0\) into the equation: \[5x - 2(0) = 10\] simplifying to \(5x = 10\). Solving for \(x\), we divide both sides by 5 to find \(x = 2\). Thus, the x-intercept is \((2, 0)\).

Step 3: Find the Y-Intercept

The y-intercept occurs when \(x = 0\). Substitute \(x = 0\) into the equation: \[5(0) - 2y = 10\] which simplifies to \(-2y = 10\). Solve for \(y\) by dividing both sides by \(-2\), finding \(y = -5\). Thus, the y-intercept is \((0, -5)\).

Step 4: Calculate the Slope

The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope. Rewrite the given equation in this form: \[5x - 2y = 10\] becomes \[2y = 5x - 10\] by moving terms. Divide through by 2: \[y = \frac{5}{2}x - 5\].Thus, the slope \(m\) is \(\frac{5}{2}\).

Key Concepts

Understanding the X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis. This means that at this point, the y-value is zero. To find the x-intercept from an equation, you set the y variable to 0 and solve for x.

For instance, take the equation of the line in the standard form: \(5x - 2y = 10\). To find the x-intercept, plug \(y = 0\) into the equation:

• \(5x - 2(0) = 10\)
• This simplifies to \(5x = 10\)

Solving for x, divide both sides by 5, giving you \(x = 2\). Therefore, the x-intercept is \((2, 0)\).

This simple step of setting y to zero and solving for x gives you the exact point on the x-axis where the line intersects. Understanding this concept helps in graphing and analyzing lines effectively.

Exploring the Y-Intercept

The y-intercept of a line is where it crosses the y-axis, which occurs at the x-value of zero. In other words, it's the point where the line comes into contact with the vertical axis. To find this point from an equation, you set x to zero and solve for y.

For the equation \(5x - 2y = 10\), set \(x = 0\) to find the y-intercept:

• \(5(0) - 2y = 10\)
• This simplifies to \(-2y = 10\)

Dividing both sides by -2 will solve for y, giving \(y = -5\). Thus, the y-intercept is \((0, -5)\).

The y-intercept is crucial for graphing because it provides a clear starting point to draw a line on a graph. It also plays a vital role in the slope-intercept form of a line equation.

Calculating the Slope

The slope of a line represents its steepness and direction. It's calculated as the "rise over run," or the change in y divided by the change in x. When dealing with the standard form of a line's equation, we'll often convert it to the slope-intercept form to find the slope directly.

Start with the equation \(5x - 2y = 10\) and aim to rewrite it in slope-intercept form \(y = mx + b\), where \(m\) represents the slope:

• Rearrange the equation to \(-2y = -5x + 10\)
• Divide every term by -2
• This results in \(y = \frac{5}{2}x - 5\)

From this equation, it's clear that the slope \(m\) is \(\frac{5}{2}\).

The slope is significant in understanding the behavior of a line. It describes how the line ascends or descends as you move along it from left to right. Comprehending the slope allows for more accurate predictions and interpretations of data modeled by linear equations.