Problem 33 - Algebra and Trigonometry

Question

\(21-48=\) Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example \(6 .\) $$ \left\\{\begin{array}{r}{3 x+2 y=8} \\ {x-2 y=0}\end{array}\right. $$

Step-by-Step Solution

Answer

The system has a single solution: \((2, 1)\).

Step 1: Solve for One Variable

Start by solving one of the equations for one variable. From the second equation, \( x - 2y = 0 \), solve for \( x \) to get \( x = 2y \).

Step 2: Substitute into the Other Equation

Substitute the expression for \( x \) from Step 1 into the first equation. The first equation \( 3x + 2y = 8 \) becomes \( 3(2y) + 2y = 8 \), which simplifies to \( 6y + 2y = 8 \).

Step 3: Simplify and Solve for Second Variable

Simplify the equation \( 6y + 2y = 8 \) to \( 8y = 8 \). Then solve for \( y \) by dividing both sides by 8, resulting in \( y = 1 \).

Step 4: Substitute Back to Find First Variable

Use the value of \( y \) found in Step 3 and substitute back into the expression found for \( x \) in Step 1. Hence, \( x = 2(1) = 2 \).

Step 5: Express the Solution as an Ordered Pair

The solution to the system of equations is \( (x, y) = (2, 1) \). This is an ordered pair that satisfies both equations.

Key Concepts

Substitution Method

The substitution method is a powerful tool for solving systems of equations. It involves expressing one variable in terms of another, which allows you to solve the system step-by-step. Here’s how it works:

• Step 1: Solve for one variable - Choose one of the given equations and solve it for one of the variables, typically the one that looks easier to isolate.
• Step 2: Substitute into the other equation - Replace the chosen variable in the second equation with the expression you found in step one. This turns a system of two variables into a solvable single-variable equation.
• Step 3: Solve for the second variable - Now that you have a single-variable equation, solve it to find the value of the second variable.

The advantage of the substitution method is that it reduces the number of variables in each equation, simplifying the process of finding the solution. Additionally, it’s particularly useful when one equation is easy to manipulate algebraically.

Ordered Pairs

When you find the values of the variables in a system of equations, you express the solutions in the form of ordered pairs, usually represented as \((x, y)\). An ordered pair neatly encapsulates the solution:

• Format: The first number corresponds to the value of the first variable (traditionally 'x'), and the second number is the value of the second variable (traditionally 'y').
• Interpretation: This representation indicates a point of intersection if you were to graph the system on the coordinate plane. It means that both variables satisfy both equations simultaneously.

When the system results in exactly one solution, you write the ordered pair reflecting these values, showing where both equations are true at the same time. If there are infinitely many solutions, you might express them as a general ordered pair based on a parameter.

Solving Linear Equations

At the core of solving systems of equations is the process of solving linear equations. These basic equations involve finding the values of unknown variables that make the equation true. Here are some key points:

• Solve by Isolation: Move terms around using basic algebraic operations (addition, subtraction, multiplication, division) to isolate the unknown variable on one side of the equation.
• Check Your Solution: Always substitute back into the original equation to verify that your solution satisfies it. This ensures accuracy and confirms that you've found the correct answer.

In our example, solving the equation \(6y + 2y = 8\) by simplifying it, we find \(y = 1\). Substituting \(y\) back lets us find \(x = 2\), confirming our solution in the ordered pair \((2, 1)\). Solving these basic equations correctly is crucial to interpreting and solving more complex systems.