91Ó°ÊÓ

In mathematics, an ordered pair is a pair of elements with a specific order. In the context of solving linear equations, it usually refers to the coordinates \((x, y)\). This is crucial because the order matters: (4, 2) is different from (2, 4).
To determine whether an ordered pair is a solution to an equation, simply plug the values of x and y into the equation and simplify. If both sides of the equation are equal after substitution, then the ordered pair is a solution. Otherwise, it isn't.
Think of it as a test to check if the given x and y coordinates satisfy the given linear equation.

The substitution method involves replacing variables in an equation with values to simplify and solve it. This is especially useful when determining if an ordered pair is a solution to a linear equation.
Follow these steps to apply the substitution method:

• Identify the given linear equation and ordered pair
• Substitute the x-value of the ordered pair into the equation
• Substitute the y-value of the ordered pair into the equation

After making these substitutions, simplify the equation to see if both sides match. If they do, then the ordered pair is a solution. If they don't, then the ordered pair is not a solution.

Simplifying equations is a vital step in solving or verifying solutions. Simplification means to make the equation as basic as possible, usually by performing arithmetic operations.
In the given example, the equation is \[ 3y + 2x = 12 \]. We substitute the values of x and y from the ordered pair (4, 2) into the equation, yielding: \[ 3(2) + 2(4) \]
By carrying out the multiplication first: \[ 6 + 8 \]
Then the addition: \[ 14 \]
We compare the simplified result \[ 14 \] with the right side of the original equation \[ 12 \]. Since \[ 14 eq 12 \], the ordered pair is not a solution.
Simplifying equations helps us clearly see if both sides of an equation balance, ensuring we understand whether the given values satisfy the equation.