Problem 4 - College Algebra Essentials

Question

determine whether the given ordered pair is a solution of the system. $$ \begin{aligned} &(8,5)\\\ &\left\\{\begin{array}{l} 5 x-4 y=20 \\ 3 y=2 x+1 \end{array}\right. \end{aligned} $$

Step-by-Step Solution

Answer

The ordered pair (8,5) is not a solution for the system of equations.

Step 1: Substitute into the first equation

The first equation is \(5x - 4y = 20\). Substitute \(x = 8\) and \(y = 5\) into the first equation. This gives \(5*8 - 4*5\), which simplifies to \(40 - 20 = 20\). Since the left hand side equals the right hand side, (8,5) is a solution for the first equation.

Step 2: Substitute into the second equation

The second equation is \(3y = 2x + 1\). Substitute \(x = 8\) and \(y = 5\) into the second equation. This gives \(3*5 = 2*8 + 1\), which simplifies to \(15 = 16 + 1\). Since the left hand side does not equal the right hand side, (8,5) is not a solution for the second equation.

Step 3: Conclusion

Even though (8,5) is a solution for the first equation, it is not a solution for the second equation. Therefore, (8,5) is not a solution for the system of equations.

Key Concepts

Ordered Pairs

An ordered pair, commonly written as (,), is a fundamental concept in algebra and represents the coordinates of a point in a two-dimensional space, often associated with a coordinate plane. The first number in the pair, , indicates the position on the horizontal axis, also known as the x-axis, whereas the second number, , represents the position on the vertical axis or y-axis. When determining if an ordered pair is a solution to a system of equations, each element of the pair is substituted into the equations. If the pair satisfies all equations in the system by rendering true statments when plugged in, then it is considered a solution to the system.

• It is important to check each equation in the system.
• If the ordered pair does not satisfy at least one equation, it is not a solution to the system.

The process of verification is to ensure that the ordered pair maintains the balance of each equation, representing a point where all equations intersect.

Substitution Method

The substitution method is one of the techniques used to find algebraic solutions to systems of equations. This method involves replacing a variable in one of the equations with an equivalent expression from another equation.

For instance, if you have two equations, and , and you solve the first equation for to get , then you can substitute into the second equation in place of . This approach often simplifies the process to a single variable equation, which can then be solved directly.

• The goal is to isolate one variable and solve for the other.
• Check if the solution satisfies both original equations to confirm its validity.

Efficiency and accuracy are the pros of the substitution method, especially for linear systems, but it can get complicated with non-linear equations or systems with multiple solutions.

Algebraic Solutions

Algebraic solutions refer to the values of variables that satisfy a given set of equations. In the context of a system of equations, an algebraic solution is a set of values for the variables that makes all the equations in the system true simultaneously.

An algebraic solution often results in an ordered pair , which signifies a point in a two-dimensional space where the graphs of the equations intersect. Essential to solving systems algebraically are methods like substitution, elimination, and graphical representation. The chosen method should provide a clear and simplified path to finding the solution.

• Maintaining the equivalence of equations through legal algebraic manipulations is crucial.
• Every step in finding an algebraic solution should be verifiable and lead back to the original equations when reversed.

Understanding the properties of equality and the algebraic behaviour of equations is essential for finding accurate and reliable algebraic solutions.