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A linear equation is a type of equation where the highest power of the variable is 1. This forms a straight line when graphed on a coordinate plane. Linear equations are fundamental in algebra and often take the simple form of \(ax + by = c\). Here, \(a\), \(b\), and \(c\) are constants, while \(x\) and \(y\) represent variables. This representation showcases how a change in one variable affects the other.

In our example problem, we have the linear equation \(4x - 3y = 1\). The goal is to determine if the point \((12, 13)\) lies on the line represented by this equation. If a point satisfies the equation, the left-hand side of the equation equals the right-hand side for those coordinates.

Linear equations are part of coordinate geometry, a branch of mathematics dedicated to visualizing geometric figures like lines and curves within a coordinate plane. Understanding these equations is crucial for topics like functions, graphing, and solving systems of equations.

The substitution method is a technique used to solve equations and check solutions, typically involving replacing a variable with a specific value. This allows you to verify if particular values satisfy a given equation. When dealing with coordinate problems, this method helps determine if a specific point lies on the line represented by an equation.

Let's apply this to our problem by substituting the given coordinates into the equation \(4x - 3y = 1\). Given the point \((12, 13)\), we first substitute \(x = 12\):

• Step 2 illustrates this by changing the equation to \(4(12) - 3y = 1\), which simplifies to \(48 - 3y = 1\).

Next, substitute \(y = 13\) into the simplified equation:

• Following Step 3, it becomes \(48 - 3(13) = 1\), further simplifying to \(48 - 39 = 1\).

Substituting both values verifies if the point satisfies the original equation. The key idea is substitution replaces variables with numbers to test solutions in a straightforward manner.

Verifying a solution involves comparing outcomes of your substitution against the expected result. For linear equations, this means checking if both sides of the equation are equal after substituting a proposed solution.

In our exercise, after following the substitution steps:

• The calculation simplified to \(48 - 39\), yielding the left side equaling \(9\).
• The original right side of the equation is \(1\).

Comparison is crucial:

• Since \(9 eq 1\), the values \((12, 13)\) do not satisfy the equation \(4x - 3y = 1\).

Solution verification ensures that we have accurately determined whether a proposed point lies on a line. This step is indispensable in validating solutions and ensuring accuracy. Missteps in this process might lead you to incorrect conclusions, making it an important part of solving any algebraic problem.

Always double-check calculations, as arithmetic errors can incorrectly suggest a point satisfies or doesn't satisfy an equation. Ensuring this accuracy builds confidence in mathematical reasoning and problem-solving strategies.