Problem 53 - Algebra 1: Concepts and Skills

Question

Determine whether the ordered pair is a solution of the equation. (Lesson 4.2) $$ 3 x+4 y=36,(4,6) $$

Step-by-Step Solution

Answer

Yes, the ordered pair (4,6) is a solution to the equation \(3x + 4y = 36\).

Step 1: Substitute the ordered pair in the equation

Place the given ordered pair (4, 6) into the equation. In an ordered pair, the first number is 'x' and the second number is 'y'. Hence, substitute 'x' with 4 and 'y' with 6 in the given equation to get: \(3(4) + 4(6) = 36\).

Step 2: Solve the equation

Calculate the left side of the equation. \(3(4) = 12\) and \(4(6) = 24\). Adding these two results gives 36. So, the equation after substitution is \(12 + 24 = 36\).

Step 3: Verify the result

Check if the left side of the equation is equal to the right side. Here, we find that both are equal (36 = 36), indicating that the ordered pair (4,6) is indeed a solution to the given equation.

Key Concepts

Ordered Pairs

An ordered pair is a fundamental element in coordinate geometry, consisting of two elements placed within parentheses, such as \( (x, y) \). In an ordered pair, the first number is the \( x \) value, also known as the abscissa, representing the horizontal position on the Cartesian plane; the second number is the \( y \) value, or the ordinate, representing the vertical position. \( (4, 6) \) is an example of an ordered pair where \( x = 4 \) and \( y = 6 \) refer to a specific point on the plane.

To determine if this pair is a solution to an equation such as \( 3x + 4y = 36 \) means to assess whether substituting these \( x \) and \( y \) values satisfies the equation. The process entails replacing each variable with corresponding values from the pair and performing the arithmetic operations to check if both sides of the equation balance.

Substitution Method

The substitution method is a key algebraic technique for solving equations, particularly useful when dealing with systems of equations or verifying solutions, as in our exercise. The process involves substituting variables with given numbers or expressions, simplifying, and solving for the remaining variables.

For instance, in confirming whether \( (4, 6) \) is a solution to \( 3x + 4y = 36 \) as done in the problem, we replace \( x \) with 4 and \( y \) with 6, and then calculate the result to see if the identity holds true. If the two sides of the equation match after substitution, the pair is indeed a valid solution. It is a direct method that can be applied to any equation that simplifies to determine the truth value of the proposed solution.

Algebraic Solutions

Algebraic solutions are the results obtained by solving an equation or a system of equations using algebraic methods such as substitution, elimination, or graphing. These solutions can be numbers, ordered pairs, or even more complex structures depending on the equation being solved.

The essence of finding algebraic solutions is rooted in the principle that both sides of an equation represent the same quantity. When an ordered pair satisfies the equation by making both sides equal after the substitution of its values for the variables, it is classified as an algebraic solution to the equation.

In this context, respectively substituting \( 4 \) for \( x \) and \( 6 \) for \( y \) accurately renders the equation \( 3x + 4y = 36 \) to an equivalent expression \( 36 = 36 \) after calculation, confirming that \( (4, 6) \) is a valid algebraic solution.