Problem 80 - Algebra 1

Question

Decide whether the ordered pair is a solution of the system. $$ \begin{aligned} &x-5 y=9\\\ &3 x+y=11 \quad(1,-4) \end{aligned} $$

Step-by-Step Solution

Answer

The ordered pair (1, -4) is not a solution of the system.

Step 1: Substitution

Substitute the ordered pair (1, -4) into both equations for variables x and y. For the first equation this results in: \(1 - 5*(-4) = 9\). For the second equation it results in: \(3*1 + -4 = 11\)

Step 2: Evaluation

Evaluate each equation to see if they are true. The first equation becomes: \(1 + 20 = 9\) which simplifies to \(21 = 9\). The second equation becomes: \(3 - 4 = 11\) which simplifies to \(-1 = 11\).

Step 3: Conclusion

Since neither of the simplified equations is true, we can conclude that the ordered pair (1, -4) is not a solution to the system of equations.

Key Concepts

Ordered Pairs

In mathematics, ordered pairs are fundamental to understanding relationships between different coordinates or variables. An ordered pair consists of two elements presented in a specific sequence, typically noted as \(x, y\). The first element represents the horizontal position (x-coordinate), while the second represents the vertical position (y-coordinate).
This concept is crucial when solving systems of equations, as it allows us to easily identify and test specific solutions. For instance, if we want to determine whether \(1, -4\) is a solution, we simply substitute these values into the equations and check the results. Remember, the order is important, so \(x\) must always be substituted first, followed by \(y\).

Solution Verification

Solution verification is a critical step in ensuring the correctness of a potential solution for a system of equations. After substituting an ordered pair into each equation of the system, the next step is to evaluate whether the resulting expressions are true.

• For the system in question, we substitute \(x = 1\) and \(y = -4\) into both equations.
• The first equation is transformed from \(x - 5y = 9\) to \(1 - 5(-4) = 9\), resulting in \(1 + 20 = 9\), which simplifies incorrectly to \(21 = 9\).
• Similarly, the second equation becomes \(3x + y = 11\) transforming to \(3(1) + (-4) = 11\), simplifying incorrectly to \(-1 = 11\).

Both statements are false, showing that the ordered pair does not satisfy the system of equations. Solution verification helps prevent errors and confirms the accuracy of results.

Substitution Method

The substitution method is a straightforward and effective technique used in solving systems of equations. It involves replacing variables with values or expressions to simplify solving the system.
Here's a snapshot of how you use it:

• Pick one of the equations from the system to solve for one variable in terms of the other. Suppose one equation is easier to manipulate than the other, start there.
• Substitute this expression into the other equation. This reduces the problem to one variable, making it easier to solve.
• Once you solve the equation, use the result to find the value of the other variable by substituting it back into one of the original equations.

In the exercise, we substituted the provided pair \(1, -4\) directly into the equations to check if these values make both statements simultaneously true. The wrong simplifications showed that this was not the solution, which means you would need to try other pairs to potentially find a solution.