Problem 17 - Algebra 1: Concepts and Skills

Question

Use the substitution method to solve the linear system. $$ \begin{aligned} &s=t+4\\\ &2 t+s=19 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system is \(s = 9\) and \(t = 5\)

Step 1: Substitute the expression of 's' from the first equation into the second equation

Substitute \(s= t + 4\) into \(2t + s = 19\), which gives us \(2t + t + 4 = 19\)

Step 2: Simplify and solve for 't'

Combine like terms to get \(3t + 4 = 19\). To isolate 't', subtract 4 from both sides of the equation, \(3t = 15\). Then divide the equation by 3 to solve for 't', we get \(t = 5\)

Step 3: Substitute 't' into the first equation and solve for 's'

Now that we have the value of 't', we can substitute it into \(s = t + 4\), giving us \(s = 5 + 4 = 9\)

Key Concepts

Understanding Linear Systems

A linear system consists of two or more linear equations that are related by having the same set of variables. These equations represent straight lines in a coordinate plane. When we solve these systems, we are searching for the point of intersection between the lines, which translates to finding the values of the variables that make all equations true.
A linear system typically appears in the form:

• Equation 1: \(y = mx + c\)
• Equation 2: \(y = nx + d\)

Where \(m\) and \(n\) denote the slopes of the respective lines, and \(c\) and \(d\) denote the y-intercepts. Solving the system gives us a pair \(x, y\) that satisfies both equations. In this case, the system consists of equations with variables \(s\) and \(t\):

• \(s = t + 4\)
• \(2t + s = 19\)

These variables 's' and 't', describe relationships and constraints within the system.

Steps to Solve Equations through Substitution

Solving equations using the substitution method involves a clear procedure where one of the equations is manipulated to express one variable in terms of the other. This expression is then substituted into the other equation to solve for the variables step-by-step.
Here is a simplified approach:

• Identify one equation where a variable can be easily isolated. From the given system, \(s = t + 4\) is already set up to isolate \(s\).
• Substitute this expression into the other equation: Replace \(s\) in \(2t + s = 19\) with \(t + 4\), resulting in \(2t + (t + 4) = 19\).
• Solve the substituted equation by isolating the remaining variable. Here, combine like terms to get \(3t + 4 = 19\), then solve for \(t\) by successive subtraction and division, leading to \(t = 5\).
• Use the found value to solve for the other variable. Substitute \(t = 5\) back into \(s = t + 4\) to find \(s = 9\).

This method allows you to systematically reduce the system to find the value of each variable, ensuring accuracy and consistency. You are effectively solving one equation to solve another, limiting the complexity of handling multiple variables at once.

Algebraic Substitution Method

The algebraic substitution method is a powerful tool in solving linear systems, especially those with two variables. It provides a strategic pathway for reducing two equations into a simpler form focused on one variable.
This method is ideal for:

• Systems where one equation is already neatly solved for one variable, making substitution straightforward.
• Students who prefer a step-by-step methodical approach, as each step builds logically on the previous one.
• Demonstrating a clear relationship between the variables involved.

In the exercise, starting with \(s = t + 4\) allowed us to easily replace \(s\) in the second equation. This clarity is the core advantage of using substitution—it simplifies the process into manageable steps. With practice, the method becomes intuitive, reinforcing the fundamental principles of algebra as each step adheres to logical transformations and arithmetic rules. By becoming comfortable with substitution, students can tackle more complex systems with confidence.