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Understanding how to substitute values into equations is fundamental for solving algebraic problems. In the context of graphs and functions, when we are given a point, like (6,2), we are actually dealing with a set of coordinates. The first number represents the x-coordinate, while the second number is the y-coordinate.

Now, to verify if this point lies on the graph of an equation, such as \(3y + x = 12\), we must replace \(x\) with 6 and \(y\) with 2. The process involves plugging these coordinates into the equation:\[3(2) + 6 = 12\]After simplification, we obtain \(6 + 6 = 12\), which verifies that the point (6,2) satisfies the equation. This substitution technique is not limited to verifying points; it's a cornerstone method used to solve various types of algebraic equations.

Linear equations form the backbone of algebra and represent relationships where variables change at a constant rate. They are equations of the first degree, meaning that the variable(s) are not raised to any power other than 1.

A standard form for a linear equation in two variables, x and y, looks like \(Ax + By = C\), where A, B, and C are constants. To find a solution to a linear equation means determining a set of values for the variables that make the equation true. In our case, we are looking at the equation \(3y + x = 12\), which is already a solvable linear equation.

To solve for y when x is known (or vice versa), we rearrange the terms: \[y = \frac{12 - x}{3}\]Finding a solution involves substituting a value for x and calculating y, or substituting a value for y and calculating x. Any point that satisfies the equation represents a solution and is a point on the graph of the equation.

Verification is a crucial step in mathematics to ensure our solutions are accurate. To verify a point in the context of an equation means to prove whether the point lies on the line or curve that the equation represents in a graph.

To verify a point, we substitute the x and y coordinates into the equation and simplify the expression. If both sides of the equation balance — they produce the same number — then the point is verified and it lies on the graph. In case of our example, by substituting (6, 2) into \(3y + x = 12\), we get \(12 = 12\), which confirms our point lies on the line represented by the equation.

Verification is not only a step to confirm a solution but also helps us understand the relationship represented by the equation. Each verified point serves as a puzzle piece that visually builds up the graph associated with the equation.