Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the system. $$ \left\\{\begin{array}{l} 2 y-5 x=0 \\ 3 y+4 x=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is (x, y) = (0, 0).

Step 1: Write Down the System of Equations

The system of equations we need to solve is as follows:\[ \begin{align*} 2y - 5x &= 0 \ 3y + 4x &= 0 \end{align*} \]

Step 2: Solve the First Equation for One Variable

From the first equation, \(2y - 5x = 0\), solve for \(y\):\[ 2y = 5x \] Divide both sides by 2:\[ y = \frac{5}{2}x \]

Step 3: Substitute into the Second Equation

Substitute \(y = \frac{5}{2}x\) from Step 2 into the second equation:\[ 3\left(\frac{5}{2}x\right) + 4x = 0 \]

Step 4: Simplify the Second Equation

Distribute the 3 into the expression:\[ \frac{15}{2}x + 4x = 0 \]Convert 4x to have a common denominator:\[ \frac{15}{2}x + \frac{8}{2}x = 0 \]Combine like terms:\[ \frac{23}{2}x = 0 \]

Step 5: Solve for x

To solve for \(x\), set the equation:\[ x = \frac{0 \times 2}{23} \]Thus, since the right side is 0:\[ x = 0 \]

Step 6: Solve for y Using x = 0

Substitute \(x = 0\) back into the expression for \(y\):\[ y = \frac{5}{2}(0) = 0 \]

Step 7: Write the Solution

The values of both variables are \(x = 0\) and \(y = 0\). Therefore, the solution to the system of equations is \((x, y) = (0, 0)\).

Key Concepts

Linear Equations

Linear equations are mathematical statements that show the relationship between two variables, often written in the form of an equation like \( ax + by = c \). Here,

• \( a \), \( b \), and \( c \) are constants.
• \( x \) and \( y \) are variables.

In linear equations, each term is either a constant or the product of a constant and a single variable. The graph of a linear equation is always a straight line. This is why they are called "linear." The goal of solving a linear equation is to find the value(s) of the variable(s) that make the equation true.
Linear equations are used across various fields, such as physics, engineering, and in different areas of mathematics. They can be extended to more complex systems where multiple equations are considered simultaneously.

Solution Methods

Solving systems of linear equations can be approached with various methods.
Some of the most common solution methods include:

• Substitution Method: Solving one equation for one variable and substituting that value into another equation.
• Elimination Method: Adding or subtracting equations to eliminate a variable and simplify the system.
• Graphical Method: Plotting the equations on a graph to identify the point where they intersect.

Each method has its advantages and can be more suitable depending on the specific problem at hand. For example, the substitution method is usually easy for smaller systems, while the elimination method handles larger systems better.

Substitution Method

The substitution method is one practical way to solve linear systems. It involves solving one of the equations for one variable. After that, the value is substituted into the other equation. In our example, we solved the first equation \( 2y - 5x = 0 \) for \( y \), yielding \( y = \frac{5}{2}x \).
Next, we substituted \( y \) into the second equation \( 3y + 4x = 0 \), by replacing \( y \) with \( \frac{5}{2}x \). This enabled us to create a single equation in terms of \( x \), which we could then solve to find \( x = 0 \). Finally, with \( x \) known, we substituted back to find \( y = 0 \).
The substitution method is especially useful for systems where one equation is simple to manipulate.

Simultaneous Equations

A system of simultaneous equations includes two or more linear equations that involve the same set of variables. The solutions to these equations are the values of the variables that satisfy all equations in the system at the same time.
In our original exercise, we dealt with the simultaneous equations:

• \( 2y - 5x = 0 \)
• \( 3y + 4x = 0 \)

These equations can be solved by finding a common solution, meaning that the values of \( x \) and \( y \) satisfy both equations. By using the substitution method, we arrived at the solution \( (x, y) = (0, 0) \). Solving simultaneous equations is key in fields such as economics, engineering, and social sciences, where complex relationships often need to be analyzed.