Verified Answer :

  • Identify the system of equations \(\{x - 6y = -20, x - 5y = -15\}\).
  • Subtract the first equation from the second to eliminate \(x\): \((x - 5y) - (x - 6y) = -15 - (-20) \implies y = 5\).
  • Substitute \(y = 5\) into one of the original equations to find \(x\): \(x - 5(5) = -15 \implies x = 10\).
  • The solution to the system is \(\boxed{x = 10, y = 5}\).

Explanation

  1. Analyzing the System of Equations

The system of equations is given by:
\[ \begin{cases} x - 6y = -20 \quad (1) \ x - 5y = -15 \quad (2) \end{cases} \]
Our objective is to solve for the variables \(x\) and \(y\) using the elimination method. In this method, we aim to manipulate the equations such that when we add or subtract them, one of the variables is eliminated, allowing us to solve for the other. Observing the coefficients of \(x\) in both equations, we see that they are both 1. This makes the elimination of \(x\) straightforward by subtracting one equation from the other.

  1. Eliminating the Variable x

To eliminate \(x\), we subtract equation (1) from equation (2):
\[ (x - 5y) - (x - 6y) = -15 - (-20) \]
Distributing the negative sign:
\[ x - 5y - x + 6y = -15 + 20 \]
Combining like terms, the \(x\) terms cancel out (\(x - x = 0\)), and we are left with:
\[ -5y + 6y = 5 \]
\[ y = 5 \]
Thus, the value of \(y\) is 5.

  1. Solving for x and Verifying the Solution

Now that we have determined \(y = 5\), we can substitute this value back into one of the original equations to find \(x\). Substituting \(y = 5\) into equation (2):
\[ x - 5(5) = -15 \]
\[ x - 25 = -15 \]
Adding 25 to both sides:
\[ x = -15 + 25 \]
\[ x = 10 \]
So, the solution to the system is \(x = 10\) and \(y = 5\). We verify this by checking in equation (1):
\[ 10 - 6(5) = 10 - 30 = -20 \]
Since both equations are satisfied, the solution is correct.

Examples

In real life, this method is useful in scenarios where two different scenarios yield linear relationships between two unknown variables. For instance, imagine a company that tracks the total cost of two different service packages based on a fixed hourly rate and a monthly maintenance fee. If Package A costs \(x - 6y = -20\) (representing a scenario with specific base cost and variable charges) and Package B costs \(x - 5y = -15\), solving this system helps determine the exact hourly rate (\(y\)) and the fixed cost (\(x\)) that makes both packages consistent with their reported totals. By identifying these values, businesses can ensure accurate billing and pricing models.

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