Verified Answer :

  • Subtract the second equation from the first to isolate \(y\): \((x + 4y) - (x + y) = 20 - 11 \implies 3y = 9 \implies y = 3\).
  • Substitute \(y = 3\) into the second equation: \(x + 3 = 11 \implies x = 8\).
  • Verify the values \(x = 8\) and \(y = 3\) satisfy both original equations.
  • The solution is \(x = 8, y = 3\), which is \(\boxed{x=8, y=3}\).

Explanation

  1. Analyzing the System of Equations

We are given a system of two linear equations with two unknowns, \(x\) and \(y\):
1) \(x + 4y = 20\)
2) \(x + y = 11\)
Our goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously. Since both equations are linear, we can easily solve this system using the method of elimination.

  1. Eliminating x and Solving for y

To eliminate the variable \(x\), we subtract the second equation from the first:
\[(x + 4y) - (x + y) = 20 - 11\]
Simplifying the left side, we get:
\[x - x + 4y - y = 3y\]
On the right side:
\[20 - 11 = 9\]
So, we have:
\[3y = 9\]
Dividing both sides by 3 gives:
\[y = 3\]

  1. Solving for x

Now that we have determined \(y = 3\), we substitute this value back into the second equation (\(x + y = 11\)) to find \(x\):
\[x + 3 = 11\]
Subtracting 3 from both sides:
\[x = 11 - 3\]
\[x = 8\]

  1. Verifying the Solution and Final Answer

To ensure the solution is correct, we verify it by substituting \(x = 8\) and \(y = 3\) into the first equation:
\[x + 4y = 8 + 4(3) = 8 + 12 = 20\]
The equation holds true. Substituting into the second equation:
\[x + y = 8 + 3 = 11\]
The equation also holds true. Thus, the solution is \(x = 8\) and \(y = 3\).

Examples

Solving systems of linear equations is a fundamental skill in algebra with many real-world applications. For example, if you are buying school supplies, imagine you buy a total of 11 pens and pencils, where pencils cost \$1 each and pens cost \$4 each. If you spent \$20 in total, you could model this as the system \(x + y = 11\) and \(x + 4y = 20\). Solving this system helps you determine exactly how many of each item you purchased, ensuring your budget is balanced and your shopping trip is perfectly accounted for.