- Rearrange the second equation to express \(y\) in terms of \(x\): \(y = -9x\).
- Substitute \(y = -9x\) into the first equation: \(5x - 7(-9x) = 0\), which simplifies to \(68x = 0\).
- Solve for \(x\) to get \(x = 0\), then substitute back to find \(y = -9(0) = 0\).
- The final solution is \((x, y) = \boxed{(0, 0)}\).
Explanation
- Analyzing the System of Equations
We are given a system of two linear equations in two variables:
\[ \begin{cases} 5x - 7y = 0 \ 9x + y = 0 \end{cases} \]
Our objective is to solve this system using the substitution method. This system is a homogeneous system of linear equations, which means that the point \((0, 0)\) is a potential solution. To solve it by substitution, we isolate one variable in one equation and substitute it into the other.
- Substitution and Solving for x
First, we isolate the variable \(y\) in the second equation, \(9x + y = 0\):
\[ y = -9x \]
Now, we substitute this expression for \(y\) into the first equation, \(5x - 7y = 0\):
\[ 5x - 7(-9x) = 0 \]
Simplify the equation:
\[ 5x + 63x = 0 \]
\[ 68x = 0 \]
Solving for \(x\), we divide both sides by \(68\):
\[ x = \frac{0}{68} = 0 \]
- Finding y and Verifying the Solution
Now that we have the value of \(x\), we can find the value of \(y\) by substituting \(x = 0\) back into our expression for \(y\):
\[ y = -9(0) = 0 \]
Thus, the only solution to the system is \(x = 0\) and \(y = 0\). We can verify this by plugging both values into the original equations:
- For \(5x - 7y = 0\): \(5(0) - 7(0) = 0 - 0 = 0\), which is correct.
- For \(9x + y = 0\): \(9(0) + 0 = 0 + 0 = 0\), which is also correct.
Therefore, the solution to the system is \((0, 0)\).
Examples
The substitution method is a powerful tool for balancing equations in physics and chemistry. For example, when you need to determine the equilibrium point in a chemical reaction, you might end up with a system of equations where different rates must balance. Solving this system allows you to find the exact concentrations of reagents needed to reach stability. Similar logic applies to balancing electrical circuits or determining the optimal point of intersection for two linear supply and demand curves.