Question

Decide whether the given ordered pair is a solution of the equation. $$-2 x-2 y=-6 ;(4,-1)$$

Short answer

Yes, the ordered pair (4, -1) is a solution to the equation -2x - 2y = -6.

Step-by-step solution

Write down the equation

Start by writing down the equation given, which is -2x - 2y = -6.

Substitute the ordered pair (x, y) = (4, -1) into the equation

Now substitute the values from the given ordered pair into the equation. This gives -2*(4) - 2*(-1) = -6.

Simplify the above equation

Perform the arithmetic operations to simplify the equation. This gives -8 + 2 = -6.

Verify if the equation is true or not

Check whether both sides of the above equation are equal. Here, -6 equals -6, so the equation is true.

Key concepts

Ordered Pairs

An ordered pair is a pair of numbers used to locate a point on a coordinate plane, often represented as \( (x, y) \). The first number, \( x \), is referred to as the abscissa and represents the position along the horizontal axis, while the second number, \( y \), known as the ordinate, indicates the position along the vertical axis.

In the context of solving linear equations, ordered pairs come into play when we're looking for solutions that satisfy the given equation. The pair \( (4, -1) \) in the exercise is a proposed solution that needs to be checked. If when substituting \( x \), and \( y \), into the equation, the left and right sides are equal, the ordered pair is indeed a solution.

Substitution Method

The substitution method is a straightforward technique for solving algebraic equations, especially useful when working with ordered pairs and systems of equations. It involves replacing variables with actual numerical values and then performing the required operations to solve the equation.

In our exercise, the substitution method was applied by substituting \( x = 4 \) and \( y = -1 \) into the given equation \( -2x - 2y = -6 \). This helps us determine if the ordered pair meets the condition set by the equation. By substituting first and then simplifying, we are essentially 'trying out' the numbers in the equation to find out if they make it true.

Arithmetic Operations

Arithmetic operations are the fundamental processes of addition, subtraction, multiplication, and division. They come into play heavily in algebra when we manipulate equations to either simplify them or solve for variables.

In our example, after substitution, we perform arithmetic operations to simplify the equation \( -2*(4) - 2*(-1) = -6 \). Multiplication is conducted first, followed by addition or subtraction, adhering to the order of operations commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction). Simplifying the equation \( -8 + 2 \), resulting in \( -6 \) confirms that both sides equal, and thus the ordered pair provided is indeed the correct solution to the linear equation.