Problem 45 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of each equation. $$ -2 x+5 y=20 $$

Step-by-Step Solution

Answer

x-intercept: (-10, 0); y-intercept: (0, 4).

Step 1: Find the x-intercept

To find the x-intercept, set the value of \( y \) to 0 and solve for \( x \). Start with the equation:\[-2x + 5(0) = 20\]This simplifies to:\[-2x = 20\]Now, divide both sides by -2:\[x = \frac{20}{-2} = -10\]Thus, the x-intercept is \((-10, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, set the value of \( x \) to 0 and solve for \( y \). Start with the equation:\[-2(0) + 5y = 20\]This simplifies to:\[5y = 20\]Now, divide both sides by 5:\[y = \frac{20}{5} = 4\]Thus, the y-intercept is \((0, 4)\).

Key Concepts

Understanding the X-Intercept

The x-intercept is a crucial concept in understanding how an equation behaves on a graph. An x-intercept is where a line crosses the x-axis, meaning the value of y at this point is always zero. For instance, consider the equation \(-2x + 5y = 20\). To find the x-intercept, we set \(y = 0\) and solve for \(x\).

Here's how to do this step-by-step:

• Start with the original equation: \(-2x + 5(0) = 20\)
• Since \(5(0)\) equals zero, the equation simplifies to \(-2x = 20\).
• Next, divide both sides by -2 to isolate \(x\): \(x = \frac{20}{-2} = -10\).

Thus, the x-intercept is the point \((-10, 0)\).

Understanding x-intercepts helps in sketching the graph of the equation, providing a critical intersection point with the x-axis.

Understanding the Y-Intercept

Just like the x-intercept, the y-intercept is a point where the graph of an equation crosses the y-axis. Here, the value of x is zero. To find the y-intercept of the equation \(-2x + 5y = 20\), we set \(x = 0\) and then solve for \(y\).

Here's how to identify the y-intercept:

• Begin with the equation: \(-2(0) + 5y = 20\).
• With \(-2(0)\) being zero, the equation simplifies to \(5y = 20\).
• Divide both sides by 5 to solve for \(y\): \(y = \frac{20}{5} = 4\).

Thus, the y-intercept is \((0, 4)\).

Y-intercepts are important as they indicate where the graph crosses the y-axis, providing another key point for plotting linear equations.

The Basics of Linear Equations

Linear equations represent straight lines when plotted on a graph. A typical linear equation in two variables can be written in the form \(Ax + By = C\). In our given equation, \(-2x + 5y = 20\), each term represents parts of the line's characteristics.

Here are some key insights about linear equations:

• The coefficients of \(x\) and \(y\), which are \(-2\) and \(5\) in our equation, affect the slope of the line.
• The constant term affects the position at which the line crosses the axes.
• By manipulating the equation to find intercepts, we determine specific points like \((-10, 0)\) for the x-intercept and \((0, 4)\) for the y-intercept, which help in graphing.

Linear equations are foundational in mathematics, and understanding how to extract details like intercepts is essential for solving complex problems.