91Ó°ÊÓ

Ordered pairs are a fundamental concept in algebra and coordinate geometry. An ordered pair is composed of two elements, typically (x, y), where 'x' represents the horizontal coordinate and 'y' represents the vertical coordinate. These pairs are used to describe the position of points in a two-dimensional space.
To ensure an ordered pair satisfies a given equation, you must substitute the x and y values into the equation and check if the equation holds true.

For example, in the equation \( y = -\frac{1}{2} x + 5 \), you can plug in the x-value from an ordered pair and solve for the corresponding y-value to see if they match. This process verifies whether the point (x, y) lies on the line described by the equation.

Linear equations are equations of the first degree, meaning the highest exponent of the variable(s) is one. These equations form straight lines when graphed on a coordinate plane. They can be written in various forms, with the slope-intercept form being one of the most common: \( y = mx + b \), where 'm' represents the slope and 'b' represents the y-intercept.
In the problem provided, the equation is \( y = -\frac{1}{2} x + 5 \). Here:

• The slope (m) is -1/2, indicating the line decreases by 1 unit on the y-axis for every 2 units it increases on the x-axis.
• The y-intercept (b) is 5, meaning the line crosses the y-axis at \( y = 5 \).

Linear equations can be solved graphically or algebraically by substituting known values of x or y to find the unknowns.

The substitution method is a technique used to solve systems of equations or to find unknown values in a single equation. It involves substituting one variable with its corresponding value or expression from another equation or context.

In this exercise, substitution is applied to complete ordered pairs as follows:

• For \(-6, y\): Substitute \( x = -6 \) into the equation \( y = -\frac{1}{2} (-6) + 5 \) and solve for \( y \).


• For \( x, 4\): Substitute \(-4 \) into the equation and solve for \( x \).


• For \ (3, y)\: Substitute (3) into the equation and solve for \( y. \)

This method can simplify the process of finding missing values and verifying ordered pairs in various algebraic contexts. It's particularly useful for students to grasp the relationship between x and y more clearly.