Problem 6 - Algebra and Trigonometry

Question

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{aligned} 12 x+15 y &=-18 \\ 2 x+\frac{5}{2} y &=-3 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has infinitely many solutions.

Step 1: Write Both Equations in Slope-Intercept Form

To graph the equations, first convert each into slope-intercept form ( y = mx + b). For the first equation: \[12x+15y=-18\]Rearranging, we get:\[15y = -12x - 18\]Divide every term by 15 to solve for y:\[y = -\frac{4}{5}x - \frac{6}{5}\]For the second equation:\[2x+\frac{5}{2}y=-3\]Rearranging, we get:\[\frac{5}{2}y = -2x - 3\]Multiplying every term by \(\frac{2}{5}\) to solve for y, we get:\[y = -\frac{4}{5}x - \frac{6}{5}\]

Step 2: Recognize the Equations are the Same

Notice that both equations simplify to the same line:\[y = -\frac{4}{5}x - \frac{6}{5}\]Since both lines are identical, every point on the line is a solution to the system.

Step 3: Determine the Number of Solutions

Since the graphs for both equations are identical, the system has infinitely many solutions. This means any coordinate pair that satisfies the equation \(y = -\frac{4}{5}x - \frac{6}{5}\) is a solution to the system.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a method of writing a linear equation in a way that is easy to graph. The general structure of this form is \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, where the line crosses the y-axis.

This form is highly useful for understanding and graphing linear equations because it immediately shows the slope and y-intercept.

• The slope \(m\) indicates the steepness of the line and its direction (positive means upward, and negative means downward).
• The y-intercept \(b\) tells you exactly where the line starts in relation to the y-axis.

To convert an equation into the slope-intercept form, isolate \(y\) on one side, rearrange other terms, and ensure \(y\) is set equal to \(mx + b\). This straightforward approach simplifies the process of plotting lines on a coordinate plane and helps in quickly identifying the line's characteristics without complex calculations.

Systems of Equations

A system of equations consists of two or more equations that share the same set of variables. The goal is to find a common solution, or set of solutions, that satisfy all equations in the system. There are several methods to solve systems of equations, such as graphing, substitution, and elimination.

Graphing is a visual approach where each equation is plotted on a graph to determine the point(s) where the lines intersect.

• If the lines intersect at exactly one point, the system has one unique solution, represented by the coordinates of the intersection point.
• If the lines are parallel and never intersect, the system has no solution.
• If both lines coincide and are identical, the system has infinitely many solutions, meaning every point on the line is a solution.

Understanding systems of equations is essential in mathematics as it can be applied to solve real-world problems, such as optimizing resources or analyzing relationships between variables.

Infinitely Many Solutions

When a system of equations has infinitely many solutions, it means that both equations describe the same line. Every point on this line is a solution to the system. Instead of having one specific point of intersection, like unique solution systems, these systems yield a continuum of solutions along the coinciding lines.

In practical terms, infinitely many solutions occur when, after simplifying, both equations transform into an identical expression. This indicates that there's no unique solution because the two lines are perfectly overlapping.

• This occurs most often when simplifying the equations yields the same slope \(m\) and the same y-intercept \(b\).
• Graphically, if plotted, you'd only see one line as the second one would lie exactly on top of it.

Recognizing infinitely many solutions in a system is crucial because it implies that any number of possible intersections exist, signifying that the equations have an inherent redundancy.