Problem 38 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{aligned} 3 y &=24-9 x \\ 3 x+y &=8 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has infinitely many solutions. \ (x, y) = (x, 8-3x) \.

Step 1: Simplify the first equation

Start with the first equation: \(3y = 24 - 9x\). Divide every term by 3 to simplify it: \[ y = 8 - 3x \]

Step 2: Substitute the simplified equation into the second equation

Substitute \(y = 8 - 3x\) from the first equation into the second equation: \[ 3x + (8 - 3x) = 8 \]

Step 3: Solve for x

Combine like terms: \[ 3x + 8 - 3x = 8 \]. Simplify this to get \[ 8 = 8 \], which is always true. Thus, there are infinitely many solutions along the line \(y = 8 - 3x) \).

Step 4: Analyze the solution

Since the equation \(8 = 8\) is always true, the system of equations has infinitely many solutions. The solution set is given by \[ (x, y) = (x, 8-3x) \].

Key Concepts

systems of equations

A system of equations consists of two or more equations with the same set of variables. The objective is to find values for the variables that satisfy all equations simultaneously. For example, the system given is:

• 3y = 24 - 9x
• 3x + y = 8

In this case, we have two equations with two variables, x and y. To find the solution, we need to find values of x and y that satisfy both equations at the same time.

substitution method

The substitution method is one popular way to solve systems of equations. Here's how it works:

• First, solve one of the equations for one of the variables.
• Next, substitute this expression into the other equation.
• Solve the new equation for the remaining variable.
• Finally, use this solution to find the value of the first variable.

In our example, we start by solving the first equation for y:

• Original equation: 3y = 24 - 9x
• Divide by 3: y = 8 - 3x

Then, we substitute y into the second equation:

• 3x + (8 - 3x) = 8
• Simplify: 8 = 8

This simplification shows that the condition 8 = 8 is always true and therefore does not give a single solution but indicates infinitely many solutions.

infinitely many solutions

When a system of equations has infinitely many solutions, it means that any value of one variable matched with a corresponding value of the other variable will satisfy both equations. This generally happens when the two equations represent the same line or parallel lines that coincide.
In our example, the simplification gave us 8 = 8, which is an identity and always true. This tells us that for any value of x, there is a corresponding value of y that satisfies both equations. The solutions can be expressed as:

• (x, y) = (x, 8 - 3x)

This means all the points (x, y) that lie on the line y = 8 - 3x are solutions to the system.

linear algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. It provides a systematic framework for understanding how systems of equations behave.
When we talk about solving a system of linear equations, we can also represent it using matrix notation and use matrix operations to find solutions. However, in simpler cases like our example, basic algebraic methods like substitution suffice. Understanding linear algebra concepts can give deeper insights and more powerful tools for solving larger and more complex systems of equations.
Through this approach, students can learn not just to solve equations but also understand the geometric interpretations, such as solutions lying on lines, planes, or any n-dimensional subspaces, depending on the number of variables and equations involved.