Problem 16 - Algebra 2

Question

Solve each system of equations by using substitution. \(3 s+2 t=-3\) \(s+\frac{1}{3} t=-4\)

Step-by-Step Solution

Answer

The solution is \(s = -7\) and \(t = 9\).

Step 1: Solve One Equation for One Variable

Start by solving one of the equations for one of the variables. Pick the easier equation for substitution. Here, we'll solve the second equation for variable \(s\). Equation: \(s + \frac{1}{3} t = -4\) Subtract \(\frac{1}{3} t\) from both sides:\(s = -4 - \frac{1}{3} t\).

Step 2: Substitute into the Other Equation

Substitute the expression derived for \(s\) from the second equation into the first equation.Original first equation:\(3s + 2t = -3\) Substitute \(s = -4 - \frac{1}{3} t\) into the first equation:\(3(-4 - \frac{1}{3} t) + 2t = -3\).

Step 3: Simplify and Solve for t

Distribute and simplify the equation:\(3(-4) - 3(\frac{1}{3}t) + 2t = -3\)\(-12 - t + 2t = -3\)Combine like terms:\(-12 + t = -3\)Add 12 to both sides:\(t = 9\).

Step 4: Solve for s

Now that we know \(t = 9\), substitute back into the equation for \(s\):\(s = -4 - \frac{1}{3}t\)\(s = -4 - \frac{1}{3} \times 9\)\(s = -4 - 3\)\(s = -7\).

Step 5: Write the Solution

Thus, the solution to the system of equations is \(s = -7\) and \(t = 9\). The ordered pair representing this solution is \((-7, 9)\).

Key Concepts

Substitution Method

The substitution method is a powerful tool used to solve systems of equations involving two or more variables. It involves expressing one variable in terms of another from one of the equations, making it easier to substitute and solve the system.
This method is particularly useful when one of the equations is simpler or already isolated.

• Start by selecting an equation with one variable that's easier to isolate.
• Express that variable in terms of the other variable.
• Substitute this expression into the remaining equation to simplify.
• Solve for the remaining variable, and then use this value to find the other variable in the system.

Because the substitution method breaks down complex problems into smaller and more manageable parts, it's often considered one of the most intuitive methods for solving linear systems.

Linear Equations

Linear equations are equations that form a straight line when graphed on a coordinate plane. They usually appear in the format of some linear combination of variables equal to a number. In these equations, each variable is raised only to the first power, and no products of variables appear.
An example of a linear equation is:

• In the equation from our problem, the first equation is: \(3s + 2t = -3\).
• The second equation is: \(s + \frac{1}{3}t = -4\).

Linear equations are the foundation of algebraic problems, helping students learn the basics of graphing, slope, and systems of equations.Through practice, solving linear equations helps students develop skills to recognize patterns and relationships between different algebraic elements.

Algebraic Solution Steps

Understanding the algebraic steps involved in solving a system is crucial for mastering algebra. The solution involves a sequential approach to simplify and solve equations one step at a time:

• Initially, isolate one variable in one of the equations. This simplification paves the way for easier substitution into the other equation.
  In our example, we solve for \(s\) in the second equation.
• Once the expression is substituted, the next step is to simplify this new equation. This step typically involves combining like terms and performing simple arithmetic.
• This leads to finding the value of one of the unknown variables.
• Finally, with one variable known, substitute it back into any earlier derived expression to find the value of the remaining variable.

The algebraic solution steps method provides a logical framework to tackle problems systematically, ensuring each part of the puzzle is addressed in sequence before moving onto the next.