Problem 43 - Algebra 1

Question

Solve the linear system. $$ \begin{aligned} &3 x+9 y=1\\\ &2 x+3 y=\frac{2}{3} \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 1/3\) and \(y = 0\).

Step 1 - Simplify the equations

The equations \(3x + 9y = 1\) and \(2x + 3y = 2/3\) can be simplified by their common divisors. The simplified system of equation is: \[\begin{cases} x + 3y = 1/3\ 2x + 3y = 2/3 \end{cases} \]

Step 2 - Subtraction

Subtract the first equation from the second equation. This results in \( x = 1/3 \)

Step 3 - Subtitution

Substitute \( x = 1/3 \) in the first equation: \(1/3 + 3y = 1/3\), which simplifies to \(y = 0\)

Step 4 - Summary of the solution

So the solution of the system is \(x = 1/3, y = 0\)

Key Concepts

Linear Equations

A linear equation is a mathematical expression that forms a straight line when graphed. These equations typically involve variables to the first power and are essential in understanding relationships between different quantities. In the context of our exercise, we deal with a system of two linear equations.

• Each equation represents a line on a graph where the solution is the point that both lines intersect.
• In our example, the equations are given as:

\[3x + 9y = 1\]\[2x + 3y = \frac{2}{3}\]Understanding the basic properties of linear equations helps in identifying solutions. When equations are linear, they find applications in a wide range of fields including economics, physics, and engineering. Their simplicity allows for fundamental concepts like slope and intercept to be easily understood, providing valuable insights into linear relationships.

Substitution Method

The substitution method is a technique for solving systems of equations. It's used to find the exact point where two lines intersect, representing the solution to the system. Here's how it generally works:

• You solve one of the equations for one of the variables to express it as a function of the other variable.
• Then, substitute this expression into the other equation.
• This substitution reduces the system into a single equation with one variable.

In our exercise, after simplifying the first equation to \(x + 3y = \frac{1}{3}\), we can solve for \(x\) using the second method, which quickly yields \(x = \frac{1}{3}\). After finding \(x\), the solution for \(y\) can be substituted back into either equation to find \(y\), yielding \(y = 0\). Thus, the substitution method streamlines the solving process by eliminating one variable at a time.

Simplifying Equations

Simplifying equations is often the first step in solving linear systems. It makes the equations easier to work with, so you can more efficiently find their solutions.

• Involves reducing coefficients or numbers within equations to smaller or simpler forms.
• This can be done by dividing terms by a common factor, allowing for more manageable numbers.

In the given exercise, starting with the system:\[3x + 9y = 1\]\[2x + 3y = \frac{2}{3}\]Both equations can be simplified. For instance, notice both terms of the first equation can be divided by 3. This simplifies our equations to:\[x + 3y = \frac{1}{3}\]\[2x + 3y = \frac{2}{3}\]By reducing complex terms, simplification makes it easier to apply other solution techniques like substitution and is a useful skill in making complex problems more approachable.