Problem 20 - Precalculus: Mathematics for Calculus

Question

Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether 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 as both equations describe the same line.

Step 1: Convert the Equations to Slope-Intercept Form

To graph the system of equations, we first convert each equation to slope-intercept form, which is \(y = mx + b\).For the first equation, \(12x + 15y = -18\):- Subtract \(12x\) from both sides: \(15y = -12x - 18\).- Divide each term by 15: \(y = -\frac{4}{5}x - \frac{6}{5}\).For the second equation, \(2x + \frac{5}{2}y = -3\):- Subtract \(2x\) from both sides: \(\frac{5}{2}y = -2x - 3\).- Divide each term by \(\frac{5}{2}\) which is equivalent to multiplying by \(\frac{2}{5}\): \(y = -\frac{4}{5}x - \frac{6}{5}\).

Step 2: Compare the Equations

Now we have both equations in slope-intercept form:1. \(y = -\frac{4}{5}x - \frac{6}{5}\)2. \(y = -\frac{4}{5}x - \frac{6}{5}\)Observe that both equations have identical slopes and y-intercepts. This means both lines are exactly the same.

Step 3: Determine the Nature of the Solutions from the Graph

Since both equations represent the same line when graphed, they will overlap completely. This indicates that the system does not have a single intersection point but rather infinite points in common. Graphically, this shows that there are infinitely many solutions because the two lines are coincident, meaning they lie on top of each other throughout.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves plotting lines on the coordinate plane. This technique visually represents equations to examine relationships between variables and discover where lines intersect. To begin graphing an equation, it must first be converted to a form that is easy to plot, such as the slope-intercept form: \(y = mx + b\).
To graph, start by identifying the y-intercept, which is the point \((0, b)\). Mark this point on the y-axis. Next, use the slope \(m\), which is expressed as a fraction, \(\frac{rise}{run}\). This tells you how to move from the y-intercept. If the slope is \(\frac{3}{2}\), move up 3 units and right 2 units to plot a new point. Draw a line through your points to complete the graphing.
Graphing helps visualize solutions of systems of equations because you can easily identify intersections, which represent solutions. Intersections show shared values of x and y for both equations, meaning they satisfy both simultaneously.

Slope-Intercept Form

The slope-intercept form is one of the most popular equations used in algebra. This form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. It's favored because it clearly shows how y responds to changes in x.

• The slope \(m\) represents the "steepness" or inclination of the line. A positive slope indicates the line rises as it moves from left to right, while a negative slope means it falls.
• The y-intercept \(b\) is where the line crosses the y-axis. If \(b = 3\), then the line passes through the point \((0, 3)\).

To convert equations to slope-intercept form, solve for y. This rearrangement helps in graphing and comparing lines to determine intersections or parallelism.
The versatility of slope-intercept form assists in solving systems of equations because by equating two lines \(y\)-values, you can solve algebraically for their intersections or graphically ascertain their behavior.

Systems of Equations Solutions

A system of equations comprises two or more equations with the same variables. Finding solutions means identifying values that satisfy all equations in the set. Systems may have different types of solutions based on how lines intersect when graphed.

• One solution occurs when lines intersect at a unique point, indicating there is precisely one set of values that satisfy both equations.
• No solution is present when lines are parallel and never meet, revealing no common values for x and y that satisfy both equations simultaneously.
• Infinitely many solutions happen when equations represent the same line. They coincide entirely, so every point on the line is a solution for both equations.

Graphing helps to visually identify these scenarios. In our example, both lines coincided, showing that the system has infinitely many solutions. Knowing how to interpret these intersections is key to solving systems of equations.