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Algebraic substitution is a method used to solve equations by replacing variables with given numbers. In this exercise, we have the equation \[4x + 7y = 56\]. We need to complete the ordered pairs so each pair satisfies the equation.
The given pairs include missing values for either x or y: \((\_, 2)\),\((\_, 0)\), and\((0, \_)\).

In the first pair, we substitute y with 2. The equation becomes \[4x + 7(2) = 56\], which simplifies to \[4x + 14 = 56\]. By solving for x, we get x = 10.5. Thus, the first pair is \((10.5, 2)\).
For the second pair, \((\_, 0)\), substitute y with 0. The equation simplifies to \[4x = 56\], so x = 14. The second pair is \((14, 0)\).
For the third pair, \((0, \_)\), substitute x with 0, leading to \[0 + 7y = 56\]. Solving for y gives y = 8. Therefore, the third pair is \((0, 8)\).

Ordered pairs are pairs of numbers \((x, y)\) that represent points in the coordinate plane. Each pair shows the position of a point along the x-axis and y-axis.
In an ordered pair, the first number represents x (horizontal position), and the second number represents y (vertical position).
To determine if an ordered pair satisfies an equation, substitute the pair's values into the equation.
For example, to check if \((10.5, 2)\) satisfies \[4x + 7y = 56\], substitute x with 10.5 and y with 2. You'll find the equation holds true, thus the pair is correct.
Similarly, substituting the values of the second pair \((14, 0)\) and third pair \((0, 8)\) verifies they satisfy the equation as well. Ordered pairs are essential in representing mathematical relationships graphically.

Simplifying equations means to reduce them to their simplest form, making them easier to solve. This process involves performing operations like addition, subtraction, multiplication, and division to both sides of the equation.
Let's take the equation \[4x + 7y = 56\]. For the first pair \((x, 2)\), substituting y simplifies the equation to \[4x + 14 = 56\]. Subtracting 14 from both sides, we get \[4x = 42\].
To simplify further, divide both sides by 4: \[x = 10.5\].
Similarly, for \((x, 0)\), substituting y simplifies it to \[4x = 56\]. Dividing by 4 gives \[x = 14\].
For the pair \((0, y)\), substituting x reduces it to \[7y = 56\], which simplifies to \[y = 8\] after dividing by 7.
Simplification is crucial as it helps in finding accurate solutions without further complications.