Problem 21 - Algebra for College Students

Question

Solve each system by using the substitution method. \(\left(\begin{array}{l}u=t-2 \\ t+u=12\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is \( t = 7 \) and \( u = 5 \).

Step 1: Identify equations

We are given two equations to solve: \( u = t - 2 \) and \( t + u = 12 \). The goal is to find the values of \( t \) and \( u \) that make both equations true.

Step 2: Solve for one variable

The first equation is already solved for \( u \), \( u = t - 2 \). We can use this expression to substitute for \( u \) in the second equation.

Step 3: Substitute and simplify

Substitute \( u = t - 2 \) into the second equation: \( t + (t - 2) = 12 \). Simplify the equation: \( 2t - 2 = 12 \).

Step 4: Solve for \( t \)

Add 2 to both sides of the equation: \( 2t = 14 \). Divide both sides by 2 to solve for \( t \): \( t = 7 \).

Step 5: Solve for \( u \)

Now that we have \( t = 7 \), use the first equation \( u = t - 2 \) to find \( u \). Substitute \( t = 7 \) into the equation: \( u = 7 - 2 = 5 \).

Key Concepts

System of Equations

In algebra, a system of equations involves two or more equations that share the same variables. Each equation in the system models a linear relationship, which can intersect, be parallel, or coincide with another, depending on their graphs.
The goal is to find a solution that satisfies all equations simultaneously. In the given exercise, we are working with the following system:

• Equation 1: \( u = t - 2 \)
• Equation 2: \( t + u = 12 \)

To find solutions for a system of equations, various methods can be employed, such as graphing, elimination, or substitution. Here, we use substitution, which is ideally suited when one equation is already solved for one variable, as it is in equation 1. When a solution is found, it represents the set of values \((t, u)\) that satisfies both equations, meaning they are true for both simultaneously.

Solving Equations

Solving an equation involves manipulating it to express a variable in terms of the others or to find its value. The process requires keeping both sides of the equation balanced by performing identical operations.
For instance, when we solve the equation \( 2t - 2 = 12 \), we aim to isolate \( t \) on one side. This can be achieved through simple algebraic operations:

• Add 2 to both sides: This gives \( 2t = 14 \).
• Divide each side by 2: This results in \( t = 7 \).

By applying these operations, we've solved for \( t \). Next, the known value is plugged back into the first equation to determine \( u \). This systematic approach ensures precision, preventing mistakes and allowing clear interpretation of the solution.

Algebraic Expressions

Algebraic expressions form the foundation of equations; they are combinations of variables, numbers, and operations. Understanding how to manipulate them is key in algebra. In our example, the expression \( u = t - 2 \) is derived from isolating \( u \) on one side, making it clear how \( u \) and \( t \) relate.
When substituting back into \( t + u = 12 \), we replace \( u \) with \( t - 2 \) to form a new expression. This new equation \( t + (t - 2) = 12 \) simplifies the problem:

• Recognize like terms: Combine \( t + t \) to get \( 2t \).
• Simplify constants: Adjust by handling the constant \(-2\).

These manipulations are cornerstones of algebra, enabling students to solve for unknowns efficiently. Mastery of these skills is crucial for solving more complex systems and navigating different forms of algebraic equations.