Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Solve each system by elimination. $$\begin{aligned}&3 x+5 y=-2\\\&9 x+15 y=-6\end{aligned}$$

Step-by-Step Solution

Answer

The system has infinitely many solutions.

Step 1: Identify and Write the System of Equations

We have the system of equations: \( 3x + 5y = -2 \) and \( 9x + 15y = -6 \). The goal is to use the elimination method to find the values of \( x \) and \( y \).

Step 2: Adjust the Equations for Elimination

Notice that the second equation \( 9x + 15y = -6 \) is a multiple of the first equation \( 3x + 5y = -2 \). Specifically, the second equation can be obtained by multiplying the first equation by 3.

Step 3: Verify if the Equations Are the Same

Multiply the first equation by 3: \( 3(3x + 5y) = 3(-2) \), which results in \( 9x + 15y = -6 \). This is identical to the second equation. This means both equations represent the same line.

Step 4: Conclusion on the Solutions

Since the two equations are identical, the system has infinitely many solutions. Any point \((x, y)\) that satisfies \( 3x + 5y = -2 \) is a solution to the system.

Key Concepts

System of Equations

A system of equations is essentially a set of two or more equations with multiple variables, usually written in terms of the same set of variables. In simpler terms, you're dealing with more than one statement or condition that involves variables like \(x\) and \(y\). The aim is to find values for these variables that satisfy all the conditions simultaneously.

• Each equation in the system can represent a rule or requirement for the variables.
• Often, these equations are linear, meaning they graph out as straight lines on a coordinate plane.
• The solution to the system is the point or points where all equations intersect when plotted.

In the given exercise, the system of equations is given by two linear equations: \(3x + 5y = -2\) and \(9x + 15y = -6\). The goal is to find the values for \(x\) and \(y\) that make both equations true.

Infinite Solutions

When we solve a system of equations, one of the possible outcomes is that there are infinite solutions. This occurs when the equations are essentially the same, meaning they express the same relationship between variables. If you graph these equations, they would lie on top of each other, forming the same line.

• The exercise's system shows infinite solutions because both equations are multiples of each other.
• Mathematically, this means any point \((x, y)\) that satisfies one equation satisfies the other.

In the provided example, by multiplying the first equation by 3, we get the second equation, proving they are not two distinct lines but the same line. This is why the system doesn't have just a single solution like an intersection point. Instead, every point on the line \(3x + 5y = -2\) is a solution.

Precalculus

Precalculus is an important course in mathematics that prepares students for calculus. It covers various algebraic, geometric, and trigonometric concepts, and one key component is the study of systems of equations. Understanding systems of equations is crucial in precalculus because it lays the foundation for solving real-world problems involving multiple unknowns.

• Precalculus students learn methods like substitution and elimination to solve systems.
• The purpose is to equip students with skills to handle more complex functions and concepts in calculus.

In this context, the elimination method used in solving the system given is a vital skill acquired in precalculus. By mastering these techniques, students are better prepared for the calculus challenges ahead, which require strong algebraic manipulation skills.

Linear Equations

Linear equations are algebraic expressions that form straight lines when graphed on a coordinate plane. They typically involve constants and linear terms (like \(x\) and \(y\)) but do not include their powers higher than one.

• The general form of a linear equation in two variables is \(ax + by = c\).
• Linear equations have a constant rate of change and a constant slope.

In the given exercise, each equation in the system, \(3x + 5y = -2\) and \(9x + 15y = -6\), is linear. Solving the system by elimination involves recognizing the alignment of these linear equations. Since they represent the same line, we conclude the system has infinite solutions — a typical scenario when dealing with linear equations and their manipulations. Understanding linear equations is crucial because they form the basis for more advanced mathematical concepts and are widely applicable in various fields.