Problem 67 - College Algebra Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The rectangular coordinate system provides a geometric picture of what an equation in two variables looks like.

Step-by-Step Solution

Answer

Yes, the statement makes sense. The rectangular coordinate system does provide a geometric representation of what an equation in two variables looks like. It enables the visual interpretation of the solution set of the equation.

Step 1: Understanding the statement

When we say 'The rectangular coordinate system provides a geometric picture of what an equation in two variables looks like', it means that using the rectangular coordinate system, we can visualize the solutions of an equation in two variables.

Step 2: Understanding the rectangular coordinate system

A rectangular coordinate system is a two-dimensional graph in which each point is identified by an ordered pair of numbers (x, y). The first number corresponds to the x-coordinate and the second to the y-coordinate. It's called 'rectangular' because it forms a grid of rectangles.

Step 3: Understanding how the coordinate system represents equations

If we take any equation in two variables, for example, \(y = 2x + 1\), we can use the coordinate system to represent it. Each solution of this equation corresponds to a point on the plane, and the collection of all such points forms a line. This line gives a 'picture' of what the equation looks like.

Step 4: Concluding whether the statement makes sense

Given the explanations above, it can be concluded that the statement does make sense. The rectangular coordinate system does indeed provide a geometric representation of an equation in two variables. It allows us to visualize the set of all possible solutions to the equation.