Problem 18 - Algebra 1

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 \( t = 5 \) and \( s = 9 \).

Step 1: Substitute the Equation

Given the two equations \( s = t + 4 \) and \( 2t + s = 19 \), we can use substitution to replace \( s \) in the second equation with \( t + 4 \). This would result in a new equation: \( 2t + t + 4 = 19 \).

Step 2: Simplify and Solve for 't'

After substituting, simplify the equation to form \( 3t + 4 = 19 \). Subtract 4 from both sides to get \( 3t = 15 \). Now, divide both sides by 3 to isolate \( t \), which gives us \( t = 5 \) as a solution.

Step 3: Solve for 's'

Once the value of \( t \) is known, substitute it back into the first equation \( s = t + 4 \) to solve for \( s \). Hence the equation becomes \( s = 5 + 4 \), which simplifies to \( s = 9 \).

Key Concepts

Linear Systems

Linear systems are collections of two or more linear equations, which can often be solved simultaneously to find the values of variables that satisfy all equations at once. In our example, we have two equations: \( s = t + 4 \) and \( 2t + s = 19 \). The goal is to find values for \( s \) and \( t \) that make both equations true simultaneously.

Linear systems can appear in many real-world contexts, like budgeting (where you're distributing resources) or physics (where multiple forces are at play). Understanding the mechanics of solving these systems is crucial because it allows us to solve diverse problems efficiently.

• Linear equations form straight lines when graphed.
• Solutions to a system represent the point(s) where these lines intersect.
• Systems can have one solution, infinitely many solutions, or no solution, depending on how the lines relate to one another.

In short, linear systems are foundational in algebra and vital in problem-solving across various fields.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. In our exercise, we solve for the variables \( s \) and \( t \) using a method called 'substitution'. This is just one of many ways to solve equations, especially when dealing with linear systems.

To solve equations effectively:

• Always perform the same operation on both sides of the equation to maintain equality.
• Simplify equations step by step to isolate the variable.
• Check your solutions by plugging them back into the original equations.

In our specific example, after substitution, we simplified \( 3t + 4 = 19 \) to find \( t = 5 \). This process involves basic arithmetic operations: subtracting, dividing, and eventually substituting back to find \( s = 9 \). Each step is logical and builds toward the solution, illustrating the importance of solving equations incrementally.

Mathematical Substitution

Mathematical substitution is a strategy used to simplify equations by replacing variables with their equivalent expressions from other equations. It's particularly useful in solving linear systems, where it can reduce complexity significantly.

The substitution method involves three main steps:

• Identify one equation where a variable is easily isolated (here, \( s = t + 4 \)).
• Replace the other equation's variable (\( s \)) with this expression (\( t + 4 \)).
• Solve the altered equation to find the remaining variables' values.

In our example, after substituting \( s \) with \( t + 4 \), we were left with a simplified equation \( 3t + 4 = 19 \).

Substitution is a powerful technique because it reduces the number of variables, leading to simpler calculations and clearer solutions. After finding one variable's value, it's straightforward to substitute back and find the other.