Problem 79 - Algebra 1

Question

Decide whether the ordered pair is a solution of the system. $$\begin{aligned}&2 x+4 y=2\\\&-x+5 y=13 \quad(-3,2)\end{aligned}$$

Step-by-Step Solution

Answer

Yes, the ordered pair (-3,2) is a solution to the given system of equations.

Step 1: Substitute Values in the First Equation

Insert the values from the ordered pair (-3,2) into the first equation, resulting in the equation \(2(-3)+4(2)=2\). This simplifies to -6+8, which equals 2.

Step 2: Substitute Values in the Second Equation

Then, we substitute the values from the ordered pair (-3,2) into the second equation, resulting in the equation \(-(-3)+5(2)=13\). This simplifies to 3+10, which equates to 13.

Step 3: Conclusion of the Verification

Since the ordered pair (-3,2) satisfies both equations, it can be concluded that it is indeed a solution to the system of equations.

Key Concepts

Understanding Ordered Pairs in Systems of Equations

When dealing with systems of equations, it's important to understand the role of ordered pairs. An ordered pair is simply a pair of numbers that represents a point on a coordinate plane. For example, the ordered pair \((-3,2)\) corresponds to the point where the x-coordinate is -3 and the y-coordinate is 2. In the context of systems of equations, an ordered pair is considered a solution if it simultaneously satisfies all equations in the system. This means that when you substitute the x and y values from the ordered pair into each equation, both sides of the equation must remain equal. Identifying solutions as ordered pairs helps to visually represent the intersection of graphs of equations, enriching our understanding of where solutions lie. It's like finding a precise spot on a map that fits multiple conditions, making ordered pairs crucial for solving and verifying solutions in systems of equations.

Applying the Substitution Method

The substitution method is a key technique in solving systems of equations. This method involves substituting one part of the equation with an equivalent expression, making it easier to solve the system.In the given problem, the substitution method helps to verify whether the ordered pair \((-3,2)\) is a solution. The process includes replacing the variables x and y in both equations using the values from the ordered pair:

• For the first equation, replace \(x\) with \(-3\) and \(y\) with \(2\), simplifying the equation to determine if both sides are equal. In this case, \(2(-3) + 4(2) = -6 + 8 = 2\), which matches the right side of the equation.
• For the second equation, do the same replacement and simplify: \(-(-3) + 5(2) = 3 + 10 = 13\), which matches too.

Substitution is an excellent strategy for finding the most convenient way to simplify and solve equations, especially when equations can be directly simplified through substitutions.

The Importance of Solution Verification

Verifying the solution of a system of equations is a crucial step in ensuring accuracy. This process confirms that a proposed solution, such as an ordered pair, truly satisfies all equations involved. Once you've used methods like substitution, the next step is to check your results thoroughly. For example, to verify the solution \((-3,2)\), you:

• Substitute \(x = -3\) and \(y = 2\) into both equations of the system.
• Check if the equations hold true, both providing equal values on each side after simplification.

If both checks result in true statements, it confirms that the ordered pair maintains the balance required by the system equations.This step is essential because it discerns correct solutions from incorrect ones and ensures that any conclusions drawn from the mathematical model accurately reflect the problem's conditions. It acts as a safeguard against oversight and helps in reinforcing mathematical comprehension.