Problem 18 - Introductory Algebra for College Students

Question

Solve each system by the substitution method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}4 x+3 y=0 \\\2 x-y=0\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is { (x, y) | x = 0, y = 0 }

Step 1: Express one variable in terms of the other

Start by expressing y from the second equation. It can be rewritten as: \(y = 2x\)

Step 2: Substitute the expression for y into the first equation

Substitute \( y = 2x \) into the first equation \( 4x + 3y = 0 \). This gives the new equation \( 4x + 3(2x) = 0 \). Simplifying this equation, we get \( 10x = 0 \)

Step 3: Solve for x

To solve for x, we divide both sides of the equation by 10 to give: \( x = 0 \)

Step 4: Substitute x back to find y

Substitute \( x = 0 \) into \( y = 2x \) to find y. This gives: \( y = 2(0) = 0 \)

Step 5: Express the solution in set notation

The solution can be written in set notation as: { (x, y) | x = 0, y = 0 }

Key Concepts

Substitution Method in Systems of Equations

The substitution method is a technique for solving systems of linear equations. Imagine you're trying to find the values of two variables that satisfy both equations at the same time. Here’s how it works:

• First, solve one of the equations for one variable in terms of the other. This is like picking apart the problem, focusing on one piece at a time.
• Next, substitute this expression into the other equation. Now you're down to just one variable, which makes it easier to solve.
• Finally, once you have the value of one variable, plug it back into the expression you found earlier to get the other variable.

For our exercise, we started by expressing \( y \) from the second equation: \( y = 2x \). Then, we substituted this into the first equation, which helped transform the problem into a single-variable equation. This effective simplification is the heart of the substitution method.

Understanding Set Notation

Set notation is a way to express solutions to systems of equations concisely. It’s like writing down the answer in a way that clearly shows all the involved conditions.

• A set is a collection of elements, often numbers, that share some property.
• For systems of equations, each solution is an ordered pair \((x, y)\) that makes both equations true.
• In set notation, the solution is often written as \( \{(x, y) | \text{conditions} \} \), which translates to "the set of all \( (x, y) \) such that the conditions are satisfied."

In the exercise, the solution set is \( \{ (x, y) | x = 0, y = 0 \} \). This tells us the specific values of \( x \) and \( y \) that solve the equations, which were both zero in this case. Understanding set notation helps us express our results clearly and precisely.

Solving Linear Equations

Solving linear equations effectively means finding the values of the variables that satisfy the equation. Linear equations are often in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.

• To solve simple linear equations, isolate the variable. Use operations like addition, subtraction, multiplication, and division.
• With systems of equations, you might need strategies like the substitution or elimination method.
• The solution should be checked by plugging it back into the original equations. If both equations hold true, you've found the solution.

In our exercise’s solution, after using substitution, we got the equation \( 10x = 0 \). This simplifies by dividing both sides by 10 to find \( x = 0 \). Solving this was straightforward, and we then used this \( x \) value to find \( y \). Solving linear equations involves logic, simplicity, and sometimes a bit of arithmetic, resulting in a particular point on the plane, a unique solution.