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When solving systems of equations, the substitution method provides a straightforward way by replacing one variable with an expression from another equation. This method involves the following key steps:

• First, solve one of the equations for one variable in terms of the other variable. For example, if we have a system where one equation is \( x + y = 10 \), we can express \( x \) as \( x = 10 - y \).
• Next, substitute this expression into the other equation. In our example, this would mean plugging \( x = 10 - y \) into the second equation, \( 3.25x + 10.50y = 61.50 \). Now, we only have one variable in the equation, making it simpler to solve.
• Once you've substituted, solve the resulting equation for the remaining variable. In the exercise, this involved isolating \( y \) in the simplified equation \( 7.25y = 29.00 \), leading to \( y = 4 \).
• Finally, substitute back to find the value of the other variable. Using the value of \( y \), we find \( x = 6 \) from \( x = 10 - y \).

The substitution method is particularly useful when one of the equations is easily rearranged to express a variable. This approach simplifies solving real-world problems, like determining quantities in different purchasing scenarios.

Linear equations are mathematical expressions that model relationships involving a constant rate of change. A system of linear equations consists of two or more linear equations with two or more variables. These equations are linear because they graph as straight lines. They usually take the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.For instance, in the prom picture problem, the equations \( x + y = 10 \) and \( 3.25x + 10.50y = 61.50 \) are both linear. Each represents a line where every point on the line is a potential solution to the equation. The solution to the system, \( x = 6 \) and \( y = 4 \), is where the two lines intersect, meaning it's the only possible combination of \( x \) and \( y \) that satisfies both equations simultaneously.Linear equations are foundational in algebra due to their simplicity and ability to model real-world situations, making them essential for solving practical problems such as budgeting and planning.

Systems of equations are incredibly useful in real-world applications, as they allow us to represent and solve complex problems with multiple constraints. This exercise exemplifies a common scenario where systems of equations are used to determine quantities or pricing in purchasing decisions. In the prom picture problem, the system of linear equations models two pieces of information: the total number of pictures and the total cost of those pictures. By creating equations from these conditions, we can solve for unknown quantities. Here are some contexts where systems of equations might be used:

• Budgeting and Financial Planning: Adjusting expenses or savings based on income constraints and financial goals.
• Resource Allocation: Assigning limited resources, like time or materials, across various tasks or projects efficiently.
• Scheduling and Planning: Planning events or work schedules to meet various commitments and constraints.

Understanding how to set up and solve these equations can make these tasks more manageable. Mastery of these concepts ensures that you can efficiently navigate real-life problems requiring multi-step reasoning and clear expressions of conditions and outcomes.