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Linear equations are fundamental in algebra and serve as the building blocks for understanding more complex mathematical concepts. A linear equation is an equation of the first degree, meaning it has no variables raised to a power higher than one. These equations are typically expressed in the form:

• y = mx + b,

where m is the slope of the line and b is the y-intercept. The slope determines how steep the line is, while the y-intercept indicates where the line crosses the y-axis.

In the context of the exercise, the given equation is y = 2x + 5. Here, the slope m is 2, and the y-intercept b is 5. This means that for each unit increase in x, the value of y increases by 2 units, and the line crosses the y-axis at the point (0, 5). Understanding this equation helps in predicting and testing pairs of values to see if they satisfy the equation.

Coordinate testing is a method used to determine if an ordered pair is a solution to an equation. An ordered pair consists of two elements: an x coordinate and a y coordinate, generally written in the form (x, y). Testing if a pair satisfies a linear equation involves substituting x and y from the pair into the equation and checking for equality.

For example, with the pair (-1, 3), substitute x = -1 into the equation y = 2x + 5:

• y = 2(-1) + 5 = 3,

the calculated y matches the pair's y value, confirming it as a solution.

In contrast, for the pair (2, 1):

• y = 2(2) + 5 = 9,

the result does not match, indicating that this pair does not satisfy the equation. This technique is essential for visually representing solutions on a graph.

Algebraic substitution is a critical process in solving equations and involves replacing one expression with another equivalent one. It is often used when testing whether an ordered pair satisfies a particular equation. This method involves plugging the x or y value from the pair into the equation.

To perform substitution, identify the variable you are replacing, typically x, and substitute its value in the equation to solve for y. Using the ordered pair (-1, 3) as an example:

• Substitute x = -1 into y = 2x + 5,

leading to y = 2(-1) + 5 equals 3. The result should match the y value of the ordered pair if it satisfies the equation.

This method of substition simplifies the verification process, making it easier to understand whether the expression holds true or not for a given pair. By practicing substitution, students can develop a deeper understanding of how variables interact within equations.