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Linear equations are one of the most fundamental concepts in algebra. They are called 'linear' because they represent a straight line when graphed on a coordinate plane. A linear equation in two variables has the form:

• y + 3x = 8

In linear equations, the objective is typically to find the value of a variable that makes the equation true. In this case, we were asked to find the value of y when x has specified values. The steps involve isolating the variable you are solving for by performing arithmetic operations such as addition, subtraction, multiplication, or division.

You start by substituting the given value into the equation and then solving for the unknown variable. This process is simple, yet powerful, as it can be applied to various problems in algebra and beyond.

In mathematics, variables are symbols used to represent unknown quantities. They can change and take on different values. In the given problem, we have two variables: x and y. The equation is expressed as:

• y + 3x = 8

This shows us that y and x are interconnected. Changing the value of x will affect the value of y and vice versa. Variables allow us to create general formulas and equations that can be used to solve a wide range of specific problems. Understanding variables is crucial because they form the basis for more complex topics in mathematics, science, and engineering.

The substitution method is a way of solving equations where you replace one variable with another expression that it is equal to. This method can solve linear equations, systems of equations, and more.

In this exercise, substitution helped us solve for y when x had specific values. Here is how it works: Substitute the given x value into the equation and solve for y. For instance,

• When x = -2:
  y + 3(-2) = 8
  => y - 6 = 8
  => y = 14
• When x = 4:
  y + 3(4) = 8
  => y + 12 = 8
  => y = -4

By substituting back into the original equation, we verify the values while systematically isolating the other variable. This makes solving equations more manageable and clear. The substitution method is particularly useful for solving systems of linear equations, simplifying the process into manageable steps.