Problem 17 - Precalculus: Functions and Graphs

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: Identify the System of Equations

The given system of equations is \( 2y - 5x = 0 \) and \( 3y + 4x = 0 \). Our goal is to find the values of \( x \) and \( y \) that satisfy both equations simultaneously.

Step 2: Solve for y in Terms of x from the First Equation

Rearrange the first equation \( 2y - 5x = 0 \) to express \( y \) in terms of \( x \):\[ 2y = 5x \] \[ y = \frac{5}{2}x \]

Step 3: Substitute y in the Second Equation

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

Step 4: Simplify and Solve for x

Simplify the equation:\[ \frac{15}{2}x + 4x = 0 \]Convert \( 4x \) into a fraction:\[ \frac{15}{2}x + \frac{8}{2}x = 0 \]Combine like terms:\[ \frac{23}{2}x = 0 \]Divide both sides by \( \frac{23}{2} \):\[ x = 0 \]

Step 5: Substitute x back to Find y

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

Step 6: Verify the Solution

Verify the solution \((x, y) = (0, 0)\) satisfies both original equations:1. Substitute into the first equation: \( 2(0) - 5(0) = 0 \), which is true.2. Substitute into the second equation: \( 3(0) + 4(0) = 0 \), which is also true. Thus, both equations are satisfied.

Key Concepts

Linear Equations

Linear equations are foundational in algebra and describe a straight line when plotted on a graph. They take the standard form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants and \( x \) and \( y \) are variables. The goal when working with linear equations in a system is to find the specific values of \( x \) and \( y \) that make both equations true simultaneously.
In the given equation system:

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

Both equations must be true for the same \( x \) and \( y \). Solving systems of equations usually involves methods such as graphing, substitution, or elimination to find these solutions.

Substitution Method

The substitution method is a straightforward way to solve a system of equations. It involves rearranging one equation to express one variable in terms of the other, and then substituting this expression into the other equation. This reduces the system to a single equation with one variable.
In our problem, this approach was used as follows:

• First, the equation \( 2y - 5x = 0 \) was rearranged to express \( y \) in terms of \( x \): \( y = \frac{5}{2}x \).
• This expression was then substituted into the second equation, \( 3y + 4x = 0 \), leading to a new equation: \( 3\left(\frac{5}{2}x\right) + 4x = 0 \).

By simplifying this new equation, we isolated \( x \) and solved for it, which then allowed us to find \( y \). This method is effective because it reduces the complexity of solving two equations down to one.

Verification of Solutions

Verification is an important step to ensure that the obtained solution satisfies both original equations. This not only confirms the correctness of the solution but also reinforces understanding of how the equations work together.
In our solution, once we found \( x = 0 \) and \( y = 0 \), verification was done by substituting these values back into both initial equations:

• For the first equation: substitute \( x = 0 \) and \( y = 0 \) into \( 2y - 5x = 0 \), resulting in \( 0 = 0 \), which is true.
• For the second equation: substitute \( x = 0 \) and \( y = 0 \) into \( 3y + 4x = 0 \), yielding \( 0 = 0 \), which is also true.

Verification shows that both equations are satisfied simultaneously, confirming the solution is correct. This step should always be performed when solving systems of equations to validate your result.