Problem 24 - Precalculus: Mathematics for Calculus

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6. $$\left\\{\begin{aligned} 3 x+2 y &=0 \\ -x-2 y &=8 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution is \((4, -6)\).

Step 1: Understand the System of Equations

We have two equations in a system: \(3x + 2y = 0\) and \(-x - 2y = 8\). We need to either find a solution, show that there is no solution, or express infinitely many solutions.

Step 2: Add the Equations

To eliminate one of the variables, we add the two equations.\[3x + 2y + (-x - 2y) = 0 + 8\]Simplifying the left side, we get:\[2x = 8\]

Step 3: Solve for x

From the equation \(2x = 8\), we divide both sides by 2 to solve for \(x\).\[x = 4\]

Step 4: Substitute x Back into One Equation

Using \(x = 4\) in the first equation: \(3x + 2y = 0\). Substitute the value of \(x\):\[3(4) + 2y = 0\]\[12 + 2y = 0\]

Step 5: Solve for y

Subtract 12 from both sides:\[2y = -12\]Divide both sides by 2:\[y = -6\]

Step 6: Verify the Solution

Substitute \(x = 4\) and \(y = -6\) into the second original equation \(-x - 2y = 8\) to verify.\[-(4) - 2(-6) = 8\]Simplifying gives:\[-4 + 12 = 8\]This statement is true, so the values \(x = 4\) and \(y = -6\) satisfy both equations.

Key Concepts

Solution of Equations

A system of equations is a set of multiple equations that all share the same variables. Solving a system means finding values for the variables that satisfy all equations at the same time. This leads to three possible outcomes:

• A single solution exists.
• There are infinitely many solutions.
• No solution is possible.

In the given task, the goal is to identify which of these situations applies to the equations:\[3x + 2y = 0 \space \text{and} \space -x - 2y = 8\]This involves determining the values of \(x\) and \(y\) that solve both equations simultaneously and checking whether they lead to any of the outcomes mentioned.

Elimination Method

The elimination method is a popular technique used to solve systems of linear equations. It focuses on removing one variable so that solving for the other becomes simpler. Here's how it works in our example:

• Step 1: Align the equations in such a way that allows for easy elimination of a variable by addition or subtraction.
• Step 2: Add the two given equations:\[3x + 2y + (-x - 2y) = 0 + 8\]Notice that the terms with \(y\) cancel each other out, leaving:\[2x = 8\]
• Step 3: Solve for \(x\) by dividing both sides of the equation by 2:\[x = 4\]
• Step 4: Substitute \(x = 4\) back into any of the original equations to find \(y\). Here we use the first equation:\[3(4) + 2y = 0\]This simplifies to:\[2y = -12\]
• Step 5: Solve for \(y\) by dividing both sides by 2:\[y = -6\]

By following these steps, the elimination method allowed us to find \(x = 4\) and \(y = -6\).

Verification of Solution

Once we have the potential solution, it's crucial to verify it to ensure it's correct. Verification involves substituting the values back into each original equation to confirm they hold true. For our system:

• In the first equation, substitute \(x = 4\) and \(y = -6\):\[3(4) + 2(-6) = 0\]Simplifies to:\[12 - 12 = 0\]This is correct!
• In the second equation, substitute the same values:\[-(4) - 2(-6) = 8\]This simplifies to:\[-4 + 12 = 8\]Which is also correct!

Since the values satisfy both equations, we confirm that \(x = 4\) and \(y = -6\) are the correct solution for the system.