Verified Answer :

  • Isolate one variable from the first equation to express it in terms of the other: \(x = 7y - 5\).
  • Substitute this expression into the second equation: \(3(7y - 5) - 8y = 11\).
  • Simplify and solve for the variable \(y\), yielding \(y = 2\).
  • Substitute \(y = 2\) back into the isolated expression for \(x\) to find \(x = 9\).
  • The solution to the system is \(\boxed{{(x, y) = (9, 2)}}\).

Explanation

  1. Problem Analysis

We are given a system of two linear equations in two variables:
1) \(x - 7y = -5\)
2) \(3x - 8y = 11\)
Our objective is to solve this system using the substitution method. This method involves isolating one variable in one equation and substituting its expression into the other equation, which reduces the system to a single linear equation with only one variable. This allows us to find the value of that variable and subsequently solve for the other.

  1. Isolating a Variable

We start by isolating \(x\) in the first equation. Starting with:
\[x - 7y = -5\]
Adding \(7y\) to both sides gives:
\[x = 7y - 5\]
Now we have an expression for \(x\) in terms of \(y\).

  1. Solving for the First Variable

We substitute the expression \(x = 7y - 5\) into the second equation of our system:
\[3x - 8y = 11\]
Replacing \(x\) with \((7y - 5)\), we get:
\[3(7y - 5) - 8y = 11\]
Now, we distribute the \(3\):
\[21y - 15 - 8y = 11\]
Combine the terms containing \(y\):
\[(21y - 8y) - 15 = 11\]
\[13y - 15 = 11\]
Adding \(15\) to both sides:
\[13y = 26\]
Dividing by \(13\):
\[y = 2\]
So, the value of \(y\) is \(2\).

  1. Solving for the Second Variable

Now that we have the value of \(y = 2\), we can substitute it back into the expression we found for \(x\) in Step 2:
\[x = 7y - 5\]
\[x = 7(2) - 5\]
\[x = 14 - 5\]
\[x = 9\]
Thus, the value of \(x\) is \(9\). The solution to the system of equations is the pair \((9, 2)\).

  1. Verification and Final Result

To verify, we check our solution in the original equations:
1) For \(x - 7y = -5\):
\(9 - 7(2) = 9 - 14 = -5\). (Correct)
2) For \(3x - 8y = 11\):
\(3(9) - 8(2) = 27 - 16 = 11\). (Correct)
The solution \((x, y) = (9, 2)\) satisfies both equations. The final answer is \(\boxed{{(x, y) = (9, 2)}}\).

Examples

Systems of linear equations are used in many real-world scenarios where multiple constraints must be met simultaneously. For instance, if you are planning a small event, you might have a fixed budget for different items (like drinks and snacks) and specific quantity requirements. If snacks cost \(\$3\) each and drinks cost \(\$8\) each, and you need to spend a total of \(\$11\) on a specific quantity of items where the relationship between the quantities is constrained by \(x - 7y = -5\), solving this system allows you to determine exactly how many snacks and drinks you can purchase to satisfy both the budget and the count requirements.

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