Problem 157 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 2 x+y=3 \\ 6 x+3 y=9 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions; both equations represent the same line.

Step 1 - Solve one equation for one variable

Choose the first equation, which is simpler to solve for one variable. Equation 1: \(2x + y = 3\)Solve for \(y\):\(y = 3 - 2x\)

Step 2 - Substitute the expression into the second equation

Substitute \(y = 3 - 2x\) into the second equation.Equation 2:\(6x + 3y = 9\)Substituting \(y\):\(6x + 3(3 - 2x) = 9\)

Step 3 - Solve for the remaining variable

Expand and simplify the equation from Step 2 to solve for \(x\):\(6x + 3 \times 3 - 3 \times 2x = 9\)\(6x + 9 - 6x = 9\)Combine like terms:\(9 = 9\)Since the variables cancel out and the resulting statement is true, this means there are infinitely many solutions.

Step 4 - Interpret the result

Since the simplified equation results in a true statement (\(9 = 9\)), the system of equations is dependent, indicating infinitely many solutions. Both equations represent the same line.

Key Concepts

substitution method

The substitution method is a way to solve systems of equations. This technique involves solving one of the equations for one of the variables. Then, you substitute this expression into the other equation. The steps are straightforward:
- Solve one equation for one variable.
- Substitute this expression into the other equation.
- Simplify and solve the resulting equation.
- If you find a specific value for the variable, substitute it back to find the other variable.
This method works well when one of the equations is easy to solve for one variable.

dependent systems

A dependent system of equations occurs when both equations actually represent the same line. This means that every solution of one equation is also a solution of the other. In such cases, instead of one unique solution, you will find infinitely many solutions.
To identify a dependent system, you can rearrange the equations and compare their structure and coefficients. When you solve a dependent system, you will often find that the variables cancel out, leaving a true statement such as \(0 = 0\) or \(9 = 9\).

infinitely many solutions

In a system of equations, having infinitely many solutions means that the equations overlap exactly. This happens when the two equations are essentially the same, and every point on one line is also on the other.
When solving, if you end up with a statement like \(9 = 9\), which is always true, this indicates that the system has infinitely many solutions. This outcome suggests that you are dealing with a dependent system where both equations describe the same geometric line in a graph.

elementary algebra

Elementary algebra is the branch of mathematics that deals with solving equations and understanding how variables interact with numbers. It involves basic operations like addition, subtraction, multiplication, and division as applied to algebraic expressions.
In the context of solving systems of equations, elementary algebra techniques include:
- Isolating variables.
- Using substitution or elimination methods.
- Simplifying expressions.
- Understanding when variables cancel out.
Mastery of elementary algebra is crucial for solving more complex problems and moving on to higher-level math.