Problem 1 - Contemporary Precalculus

Question

Determine whether the given values of \(x, y,\) and z are a solution of the system of equations. $$\begin{aligned}x=-1, y &=3 \\\2 x+y &=1 \\\\-3 x+2 y &=9\end{aligned}$$

Step-by-Step Solution

Answer

Answer: Yes, the given values \(x = -1\) and \(y = 3\) form a solution to the given system of equations.

Step 1: Substitute the given values of x and y into the equations.

First, substitute \(x = -1\) and \(y = 3\) into the first equation: $$(-1) = -1$$ Then, substitute \(x = -1\) and \(y = 3\) into the second equation: $$2(-1) + 3 = 1$$ Lastly, substitute \(x = -1\) and \(y = 3\) into the third equation: $$-3(-1) + 2(3) = 9$$

Step 2: Evaluate the resulting equalities.

Now, we will evaluate each of the equalities from Step 1: 1. For the first equality, we directly get \(x = -1\). Since the given value of \(x\) is also \(-1\), this equality is true. 2. Evaluate the second equality: $$-2 + 3 = 1$$ $$1 = 1$$ Since the left-hand side is equal to the right-hand side, this equality is true as well. 3. Evaluate the third equality: $$3 + 6 = 9$$ $$9 = 9$$ Again, both sides are equal, meaning that this equality is also true.

Step 3: Conclusion.

Since all three equalities resulting from substituting the given values of \(x\) and \(y\) are true, we can conclude that the given values \(x = -1\) and \(y = 3\) form a solution to the given system of equations.

Key Concepts

Solution Verification

Solution verification is a critical step in solving a system of equations. It determines whether the proposed solution truly satisfies all the given equations. In this example, we check if the values \( x = -1 \) and \( y = 3 \) satisfy the system:

• The first equation, after substituting \( x = -1 \), naturally holds as \( -1 = -1 \), verifying that the solution is consistent with this equation.
• For the second equation, substituting \( x = -1 \) and \( y = 3 \) leads to \( 2(-1) + 3 = 1 \). Simplifying, we find \( -2 + 3 = 1 \), which results in \( 1 = 1 \), confirming the equality.
• Similarly, the third equation with these values yields \( -3(-1) + 2(3) = 9 \). Simplifying gives \( 3 + 6 = 9 \), which matches \( 9 = 9 \).

Each substituted equation balances both sides, validating that \( x = -1 \) and \( y = 3 \) is indeed the solution for the system.

Substitution Method

The substitution method is a technique for solving systems of linear equations by replacing variables with known values from one equation into another. This method allows us to simplify and solve the system step-by-step.Here, we use the values \( x = -1 \) and \( y = 3 \) and substitute them into each equation of the system. This method enables:

• Verification of the proposed solution by ensuring each equation is true when substituted with these values.
• Simplification of complex equations, making them easier to solve step-by-step.

The substitution method streamlines the problem-solving process by handling one equation at a time, confirming that the proposed solution holds for the entire system.

Linear Equations

Linear equations represent relationships with constant rates of change and result in straight lines when graphed. They are fundamental to mathematics and are often encountered in systems of equations.In our system:

• The first equation is simply \( x = -1 \), straightforward as it directly states the value of \( x \).
• The second equation, \( 2x + y = 1 \), and the third, \( -3x + 2y = 9 \), both describe linear relationships between \( x \) and \( y \).

Linear equations make it possible to easily plot and understand the interaction between variables. Solving systems of linear equations, like in this exercise, uncovers the intersecting point of the lines, which is the solution to the system. Such equations form the backbone for solving countless real-world problems by modeling various relationships.