Problem 5 - Elementary Algebra

Question

What is the geometric structure of the graph of all the solutions to the equation \(2 y+3 x=-4 ?\)

Step-by-Step Solution

Answer

Answer: The geometric structure of the graph of all the solutions to the equation \(2y + 3x = -4\) is a straight line with a slope of \(-\frac{3}{2}\) and a y-intercept of -2.

Step 1: Converting equation to slope-intercept form

To start, we will convert the given equation \(2y + 3x = -4\) into slope-intercept form (\(y = mx + b\)) where \(m\) is the slope and \(b\) is the y-intercept. Rewrite the equation as follows: \(2y = -3x - 4\) Now, divide both sides by 2 to solve for \(y\): \(y = -\frac{3}{2}x - 2\)

Step 2: Identifying the slope and y-intercept

Now that we have the equation in slope-intercept form, we can easily identify the slope and y-intercept. Comparing our equation (\(y = -\frac{3}{2}x - 2\)) with the general form (\(y = mx + b\)), we see that the slope \(m\) is \(-\frac{3}{2}\) and the y-intercept \(b\) is \(-2\).

Step 3: Describing the geometric structure

The geometric structure of the graph of the solutions to the equation is a straight line, as it is a linear equation. The line has a slope of \(-\frac{3}{2}\), meaning it will tilt downwards at an angle (negative slope) and for each unit increase in \(x\), the value of \(y\) will decrease by \(\frac{3}{2}\) units. The y-intercept is -2, so the line will cross the y-axis at the point \((0, -2)\). In conclusion, the geometric structure of the graph of all the solutions to the equation \(2y + 3x = -4\) is a straight line with a slope of \(-\frac{3}{2}\) and a y-intercept of -2.

Key Concepts

Slope-Intercept Form

Linear equations can often be best understood when written in what is called the slope-intercept form. This form is expressed as \(y = mx + b\). Here, \(m\) represents the slope and \(b\) is the y-intercept. The slope-intercept form makes it clear how the line will appear when graphed.
The slope \(m\) indicates the steepness and direction of the line. The y-intercept \(b\) tells us where the line will cross the y-axis, making it easy to graph.
To convert any linear equation to this form, you need to solve for \(y\) in terms of \(x\). For instance, from the given equation \(2y + 3x = -4\), we rearrange it into \(y = -\frac{3}{2}x - 2\).
This simple rearrangement provides a clear visual snapshot of how the line behaves. Without changing the solutions of the equation, we make the mathematical relationship more perceivable.

Geometric Structure

The concept of geometric structure in the context of equations refers to the visual appearance of solutions on a graph. For linear equations, like the one given, the structure is a straight line. Knowing the geometric structure helps us easily predict how a graph might look.
In our specific example, the line has a slope of \(-\frac{3}{2}\). This means for every unit increase in \(x\), \(y\) decreases by \(\frac{3}{2}\). The slope being negative indicates the line will tilt leftwards.
Furthermore, the y-intercept of \(-2\) shows that the line will intersect the y-axis at the point \((0, -2)\). This intercept serves as a starting point for graphing the line, ensuring the geometric structure is clear and accurate.
These elements combined allow us to understand the line's direction and starting point, providing a complete picture of the geometric structure.

Graphing Linear Equations

Graphing linear equations involves representing the solutions of an equation visually on a coordinate plane. It transforms mathematical expressions into an easy-to-understand picture.
For any equation in slope-intercept form \(y = mx + b\), graphing begins by plotting the y-intercept, which is the point where the line crosses the y-axis. In our example, it's \( (0, -2) \).
Next, the slope \(m\) guides how steep the line is and its direction. Here, the slope \(-\frac{3}{2}\) means from each point, you move down 3 units for every 2 units moved to the right, plotting each point accordingly.
These steps lead to a complete drawing of the line on the graph, revealing all potential solutions to the equation. Graphing illuminates the equation dynamically, helping you easily see the relationship between variables \(x\) and \(y\).