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The substitution method is a cornerstone of algebra, particularly useful when solving equations. In essence, it involves replacing a variable with its known value, which simplifies the equation and makes it easier to solve.

For example, consider the equation given in the exercise, where we are asked to find the value of the variable t when x is known to be 3. Following the substitution method, we insert the value of 3 for every occurrence of x in the original equation t = 21x + 6. This transforms our equation to t = 21(3) + 6, removing the unknown x and making t the only variable we need to solve for.

The method consistently provides a clear path towards the solution. It is equally beneficial in more complex scenarios, like solving systems of equations, where you can solve for one variable, then substitute its value in another equation to find additional unknowns.

Linear equations are foundational in algebra and represent relationships between variables where each term is either a constant or the product of a constant and a single variable. These equations form straight lines when graphed on a coordinate plane. The general form of a linear equation with two variables, x and y, is y = mx + b, where m stands for the slope and b represents the y-intercept.

In the given exercise, the equation t = 21x + 6 is linear because the variable t changes at a constant rate with x. There are no powers or roots of the variables involved. Solving such an equation when given a value for x entails a simple process of arithmetic operations—substitution, multiplication, and addition. By following these steps, we can find the corresponding value of t, which in this example is 69.

Algebraic expressions are combinations of variables, numbers, and operations that represent a specific value when the variables are replaced by actual numbers—much like a recipe that tells us what to combine and how to combine them to produce a certain result. These expressions do not contain equality or inequality signs, unlike equations.

The equation in our exercise contains an algebraic expression on the right side: 21x + 6. When we solve for t, we are essentially evaluating the algebraic expression for a given value of x. In simple terms, you just feed the value into the expression and follow the operations in the correct order—multiplying 21 by the given value of x (in this case, 3), and then adding 6 to the result. By understanding algebraic expressions and how to manipulate them, students are better equipped to tackle a wide range of mathematical problems.