Question

Find all solutions (if any) of the following systems of linear equations. a. \(3 x-2 y=5\) $$ \text { b. } \quad 3 x-2 y=5 $$ $$ \begin{array}{ll} -12 x+8 y=-20 & -12 x+8 y=16 \end{array} $$

Short answer

The system has infinitely many solutions, all satisfying the equation \(3x - 2y = 5\).

Step-by-step solution

Analyze and Simplify Equations

Let's first analyze the system of equations for the problemb:\[3x - 2y = 5 \]\[-12x + 8y = -20\]Observe if the second equation can be simplified or modified to be similar to the first.Divide the second equation by -4:\[(-12x + 8y = -20) \div (-4)\]This simplifies to:\[3x - 2y = 5\]This is the same as the first equation. Thus, both equations of the system can be reduced to:\[3x - 2y = 5\]

Identify the Nature of the System

The simplified system is:\[3x - 2y = 5\]\[3x - 2y = 5\]Both equations are identical, leading us to the possibility that the system has infinitely many solutions. Any point \((x, y)\) that satisfies the equation \(3x - 2y = 5\) will be a solution.

Explore Solutions to Single Equation

To find the set of all solutions, solve for one variable in terms of the other. From the first equation:\[3x = 2y + 5\]\[x = \frac{2y + 5}{3}\]Thus, any value of \(y\) will give a corresponding value of \(x\) and will be a solution to the system.

Key concepts

Solution of Equations

In solving systems of linear equations, we aim to find values for the variables that satisfy all equations in the system. Each equation represents a relationship between the variables, such as the equation from our problem:

• \(3x - 2y = 5\)

To solve this system, we analyze both equations for consistency and simplify when needed. The goal is to express one variable in terms of the other, enabling us to explore all possible solutions.
In our exercise, part b involves the equations:

• \(-12x + 8y = -20\)

By simplifying the equations, we can identify the nature of the solutions.

Identical Equations

Identical equations mean that two or more equations in a system are essentially the same after simplification. When solving system b, we found that simplifying ul>

\(-12x + 8y = -20\)

resulted in \(3x - 2y = 5\), just like the first equation. Both equations became identical, so they represent the same line or relationship.
This means any solution satisfying the first equation will also satisfy the second one. Understanding this helps us because it reduces the complexity of the system and confirms that if you solve one of these, you’ve solved them all.

Infinite Solutions

When a system of equations comprises identical equations, it indicates that there are infinite solutions. In other words, there are infinitely many pairs

• \((x, y)\)

that will satisfy the system.
For system b, since both equations simplify to \(3x - 2y = 5\), any point that lies on this line is a solution.
This means there is no unique solution—the graph of the lines overlaps entirely. Every point on that line represents a valid solution. It’s important to know this because it indicates flexibility in the conditions that need to be met, allowing for multiple valid solutions.

Simplifying Equations

Simplifying equations is a crucial step in solving linear systems. It involves reducing equations to their simplest form, making them easier to analyze and solve. For instance, in our example, by dividing the equation \(-12x + 8y = -20 \) by -4, we get:

• \(3x - 2y = 5\)

This makes both equations identical and easier to handle. Simplification can reveal whether the system has unique, infinite, or no solutions at all.
It helps identify relationships between variables clearly, sometimes converting complex systems into much simpler ones. By reducing unnecessary components, you can focus on the core problem and find solutions more effectively.