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Linear equations are a foundational concept in algebra. They represent equations involving only the first powers of the variables, meaning no variables are multiplied together or raised to any powers other than one. Each linear equation can be written in a form such as \[ ax + by = c \]where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) represent variables.
Linear equations are often used to describe relationships where there is a constant rate of change. In the context of our problem, the equation \( 3x + 7y = 48 \) illustrates a relationship between the points scored by field goals \((3x)\) and touchdowns \((7y)\). Understanding linear equations helps students connect real-world situations with mathematical representations, allowing us to model scenarios using algebraic methods.

Solving equations is the process of finding the value of variables that satisfy an equation. In any mathematical equation, finding a solution involves several key steps:

• Setting up the equation: Identify and express the given information and relationships in mathematical terms.
• Simplifying the equation: Combine like terms and perform any necessary mathematical operations.
• Isolating the variable: Use inverse operations to solve for the variable of interest.

In our case, we simplified the initial equations \( x + y = 8 \) and \( 3x + 7y = 48 \) to find the number of field goals and touchdowns. By isolating \( y \) and solving for \( x \), we worked step-by-step to find valid solutions. This methodical approach allows for confident and accurate results.

The substitution method is one approach to solving a system of equations. When employing this method, one equation is solved for one of the variables, and this expression is substituted into the other equation.
This transforms the system into a single equation with one variable, which can then be solved using algebraic techniques. Here’s how it applies to our problem:

• First, solve \( x + y = 8 \) for \( x \): \( x = 8 - y \)
• Substitute \( 8 - y \) for \( x \) in \( 3x + 7y = 48 \), transforming it into \( 3(8 - y) + 7y = 48 \). This leaves you with one equation to solve for \( y \).

The simplicity of this method makes it a preferred choice for many, as it reduces the problem to a straightforward algebra equation to solve.

The elimination method is another technique to solve systems of equations. It involves adding or subtracting equations in order to eliminate one of the variables, making it easier to solve for the remaining variable. Here's a basic guide on using this method:

• Align the equations such that corresponding variables are in line.
• Multiply one or both equations by a number that will allow the elimination of one variable when equations are added or subtracted.
• Add or subtract the equations to remove one variable, then solve for the remaining variable.

Although not used in our given problem's solution, the elimination method provides a valuable alternative, especially when substitution is complex or cumbersome.