Problem 17 - Understanding Elementary Algebra with Geometry

Question

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

Step-by-Step Solution

Answer

Both x- and y-intercepts are at (0, 0).

Step 1: Understanding the Equation

The given equation is \[ 2x = 5y \]. To find the intercepts, determine where the line crosses the x-axis and y-axis.

Step 2: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). \[ 2x = 5(0) \] \[ 2x = 0 \] \[ x = 0 \] The x-intercept is at \((0, 0)\).

Step 3: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). \[ 2(0) = 5y \] \[ 0 = 5y \] \[ y = 0 \] The y-intercept is at \((0, 0)\).

Step 4: Interpret the Results

Both the x-intercept and y-intercept of the equation \(2x = 5y\) are the same. The line crosses the origin at point \((0, 0)\).

Key Concepts

Linear Equations

A linear equation is an equation that plots a straight line on a graph. These equations typically take the form of \(Ax + By = C\) where \(A\), \(B\), and \(C\) are constants. Linear equations are fundamental in algebra and are used in various fields such as physics, economics, and engineering. In the given equation \(2x = 5y\), we are dealing with a linear relationship between \(x\) and \(y\). A crucial property of linear equations is that they yield graphs that are straight lines, which can be defined completely by two main points: the intercepts.

Solving for Intercepts

Intercepts are points where the graph of an equation crosses the x-axis and y-axis. They offer a very useful way to understand the behavior of linear functions. To find the intercepts of the equation \(2x = 5y\):

• x-intercept: Set \(y = 0\) and solve for \(x\)
• y-intercept: Set \(x = 0\) and solve for \(y\)

For the equation \(2x = 5y\):
1. Set \(y = 0\):
\(2x = 5(0)\)
\(2x = 0\)
\(x = 0\)
The x-intercept is \((0, 0)\).
2. Set \(x = 0\):
\(2(0) = 5y\)
\(0 = 5y\)
\(y = 0\)
The y-intercept is \((0, 0)\).
In this case, both intercepts occur at the same point, the origin \((0, 0)\).

Algebra

Algebra involves working with mathematical symbols to solve problems. It's an essential branch of mathematics that introduces using variables to represent numbers. In our problem, we solve the equation \(2x = 5y\) by substituting values for \(x\) and \(y\) to find the intercepts.
Key skills in algebra include:

• Manipulating equations: Adding, subtracting, multiplying, or dividing both sides of equations to isolate the variable of interest
• Substitution: Plugging in values for known variables to solve for unknowns
• Simplifying: Reducing expressions to their simplest form to make calculations easier

When we set \(y = 0\) to find the x-intercept and \(x = 0\) to find the y-intercept, we use these key algebraic principles to solve the problem step by step. Algebra helps us to systematically understand and find solutions to mathematical problems, making it a powerful tool in both simple and complex scenarios.