Problem 10 - College Algebra

Question

Determine whether the given ordered pair is a solution to the system of equations. $$ \begin{aligned} x+8 y &=43 \\ 3 x-2 y &=-1 \text { and }(3,5) \end{aligned} $$

Step-by-Step Solution

Answer

The ordered pair \((3, 5)\) is a solution to the system of equations.

Step 1: Substitute into the First Equation

Substitute the ordered pair \((3, 5)\) into the first equation \(x + 8y = 43\). Replace \(x = 3\) and \(y = 5\) to see if the equation holds true.\[3 + 8(5) = 43\]Calculate the left side:\[3 + 40 = 43\]Since both sides are equal, the ordered pair satisfies the first equation.

Step 2: Substitute into the Second Equation

Now substitute the ordered pair \((3, 5)\) into the second equation \(3x - 2y = -1\). Replace \(x = 3\) and \(y = 5\) and check for truth.\[3(3) - 2(5) = -1\]Calculate the left side:\[9 - 10 = -1\]Since the left side is equal to the right side, the ordered pair satisfies the second equation.

Step 3: Conclusion

Since the ordered pair \((3, 5)\) satisfies both equations, it is a solution to the system of equations.

Key Concepts

Understanding Ordered Pairs

Ordered pairs are a fundamental part of understanding systems of equations. An ordered pair is simply a pair of numbers written in a specific order. In this context, it's expressed in the form \((x, y)\). The first number represents the value of \(x\), and the second number represents the value of \(y\). It's important to remember that the order matters greatly; we always interpret these in the form \((x, y)\) and not \((y, x)\).
In the given problem, the ordered pair \((3, 5)\) needs to be checked to see if it satisfies two different equations. This means substituting \(3\) for \(x\) and \(5\) for \(y\) in each equation. By doing this, we can determine whether the ordered pair is a valid solution for the system of equations.

Exploring the Substitution Method

The substitution method is a practical technique used to determine if an ordered pair is a solution to a system of equations. This approach involves replacing, or "substituting," the values from the ordered pair into each equation of the system.
In our exercise, we take the ordered pair \((3,5)\) and substitute \(x = 3\) and \(y = 5\) into the first equation:

• Equation: \(x + 8y = 43\)
• Substituting: \(3 + 8(5)\)
• Calculate: \(3 + 40 = 43\)

Since the equation holds true, the pair \((3, 5)\) satisfies the first equation. We repeat this for the second equation:

• Equation: \(3x - 2y = -1\)
• Substituting: \(3(3) - 2(5)\)
• Calculate: \(9 - 10 = -1\)

Again, the equation holds true, confirming the ordered pair as a solution to both equations using the substitution method.

Solutions to Equations in Systems

In mathematics, when we talk about solutions to equations, we're referring to values that make the equation true. For systems of equations, a solution is an ordered pair that satisfies all equations in the system simultaneously. In other words, the solution makes each equation in the system balance when the values are substituted.
For the given system:

• First equation: \( x+8y=43 \)
• Second equation: \( 3x-2y=-1 \)

After substituting \((3, 5)\) into both equations, we found that both left-hand sides equal the respective right-hand sides.
This confirms that \((3, 5)\) is a solution to the system of equations, as it satisfies both criteria. Understanding this helps in accurately solving more complex equations and systems, giving assurance with a logical structure to find solutions.