Problem 45 - Introductory Algebra for College Students

Question

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{aligned} 3 x-2 y &=8 \\ x &=-2 y \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution for the system of linear equations \(3x-2y = 8\) and \(x = -2y\) is the set \(\{(2, -1)\}\)

Step 1: Verify that it is a system of linear equations

Each equation is linear, as it only contains constants and variables raised to the first degree. Indeed, these include \(3x-2y = 8\) and \(x = -2y\). So, yes, this is a system of linear equations.

Step 2: Solve by substitution

The second equation, \(x = -2y\), is already solved for x. This expression for x can be substituted into the first equation, replacing all instances of x, and then the resulting equation can be solved for y. To do this, substitute \(-2y\) for \(x\) in the first equation, obtaining: \(3(-2y) - 2y = 8\). This simplifies to \(-6y - 2y = 8\), and further to \(-8y = 8\). By dividing each side by \(-8\), we find that \(y = -1\).

Step 3: Substitute y = -1 to find x

Substitute \(y = -1\) into the equation \(x = -2y\), which was previously solved for x. Hence, \(x = -2(-1)\), which gives \(x = 2\).

Step 4: Write the final solution in set notation

The solution of the system is the set of all pairs of x and y that satisfy both equations. Since \(x = 2\) and \(y = -1\) are the only solutions, the solution set is \(\{(2, -1)\}\).

Step 5: Verify the solution

Substitute \(x = 2\) and \(y = -1\) into both of the original equations, verifying that each one is indeed true. For the first equation, we find \(3*2 - 2*-1 = 8\), which simplifies to \(6 + 2 = 8\), verifying that the first equation is true. Substituting into the second equation, we find \(2 = -2*-1\), which simplifies to \(2 = 2\), verifying the second equation is true as well. Thus, the solution satisfies the original system.

Key Concepts

Substitution Method

The substitution method is a powerful technique used to solve systems of linear equations, particularly when one of the equations is easy to solve for one variable. In this exercise, the equation \(x = -2y\) is simple to manipulate, making substitution the ideal choice. Here’s how it works:

• Choose one equation and solve for one variable. Here, the second equation is already solved for \(x\), \(x = -2y\).
• Substitute the expression for this variable into the other equation. Substituting \(-2y\) for \(x\) in the first equation \(3x - 2y = 8\), gives \(3(-2y) - 2y = 8\).
• Solve the new equation for the remaining variable. Simplify to find \(y = -1\).
• Back-substitute to find the other variable. Using \(x = -2y\), substitute \(y = -1\) to get \(x = 2\).

After solving, you have values for both \(x\) and \(y\), forming the solution to the system. This method is especially useful for its straightforward and methodical approach.

Solution Set

Once you have the values for the variables that satisfy the system of equations, it’s time to express the solution clearly. This brings us to the concept of a solution set. The solution set contains every pair of \((x, y)\) values that solve the system. For our exercise, we found that \(x = 2\) and \(y = -1\). Thus, the solution set is denoted as \[\{(2, -1)\}\]This notation succinctly communicates that the pair \((2, -1)\) satisfies both equations simultaneously. By using solution sets, we can clearly communicate to others the results of our problem-solving process, allowing them to easily verify and understand the solutions.

Set Notation

Set notation is a mathematical way of expressing a collection of unique elements, often showing solutions of equations or inequalities. In the context of solving a system of linear equations, set notation helps represent all solutions in a compact form.For the solution to this exercise, the pair \((2, -1)\) is the only pair that satisfies the given equations. In set notation, this is represented as\[\{(2, -1)\}\]Set notation is efficient and clear, quickly conveying the idea that any solution must adhere to a specific form. This concise communication form is universally understood in mathematical discussions, making it an important tool in your math toolkit.

System of Equations Verification

Verification is a crucial step that ensures that the solution obtained is correct. We go back to substitute the values of \(x\) and \(y\) into the original equations to verify their consistency.To verify the solution \((2, -1)\):

• Substitute \(x = 2\) and \(y = -1\) into the first equation: \(3(2) - 2(-1) = 8\). This simplifies to \(6 + 2 = 8\), confirming the equation is true.
• Substitute into the second equation: \(2 = -2(-1)\), simplifying to \(2 = 2\), which is also accurate.

Both verifications confirm the values provide consistent solutions across the system of linear equations. This crucial step ensures the solution is not only theoretically sound but practically valid as well.