91Ó°ÊÓ

A coordinate plane is a two-dimensional space where we can graph points and visualize relationships like linear equations. It consists of two perpendicular lines called axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Where they intersect is known as the origin, labeled (0,0).
The coordinate plane allows us to plot points using pairs of numbers called coordinates, written as (x, y). The x-value tells us how far to move horizontally from the origin, and the y-value tells us how far to move vertically.

• The x-axis is the horizontal number line.
• The y-axis is the vertical number line.
• The point where they meet is the origin (0,0).

This plane is especially useful in solving linear equations like the one in our exercise, where we plot each pair (x, y) to represent solutions. It helps us see how changing x affects y, giving a visual representation of the equation's slope and intercept.

A function table is a handy way to organize and understand the relationship between variables in a function or equation, like the one given by our linear equation: \(y = 2x - 1\). It's simply a table where you list some chosen x-values in one row and their corresponding y-values, calculated from the equation, in another.
A function table is useful for seeing patterns and making predictions. In our exercise, we made a function table using x-values of -3, -1, 0, and 2, then calculated the y-values:

• For \(x = -3\), \(y = -7\)
• For \(x = -1\), \(y = -3\)
• For \(x = 0\), \(y = -1\)
• For \(x = 2\), \(y = 3\)

This table helps us understand how the function behaves, showing a clear pattern where increasing x by one unit results in a two-unit increase in y, minus 1, consistent with the formula \(y = 2x - 1\).

Algebraic substitution is a fundamental technique used to solve equations by replacing variables with numbers. It's like a "plug and play" approach where you substitute a specific value into the equation to calculate another variable.
In our exercise, we used substitution to calculate y-values for given x-values. For example, considering the equation \(y = 2x - 1\):

• To find y when \(x = -3\), we substitute -3 for x: \(y = 2(-3) - 1 = -7\).
• Similarly, for \(x = 0\), substitute 0: \(y = 2(0) - 1 = -1\).

This technique is crucial for working with linear equations, as it allows us to easily compute solutions and verify our results. Substitution not only simplifies calculations but also reinforces understanding of the relationship between the variables in an equation.