Verified Answer :

  • Multiply the first equation by 4 and the second by 3 to equate the coefficients of \(y\): \(32x + 36y = 512\) and \(15x + 36y = 393\).
  • Subtract the modified equations to isolate \(x\): \(17x = 119\), which gives \(x = 7\).
  • Substitute \(x = 7\) into the first equation to solve for \(y\): \(8(7) + 9y = 128 \implies 9y = 72 \implies y = 8\).
  • The solution to the system is \(\boxed{x=7, y=8}\).

Explanation

  1. Analyzing the System of Equations

We are tasked with finding the values of \(x\) and \(y\) that satisfy the following system of linear equations:
\[ \begin{cases} 8x + 9y = 128 \ 5x + 12y = 131 \end{cases} \]
Our objective is to solve for the variables \(x\) and \(y\) simultaneously. A standard approach to solving such a system is the method of elimination, which involves manipulating the equations to eliminate one of the variables, thereby allowing us to solve for the other.

  1. Eliminating the Variable y

To use the elimination method, we look for a common multiple of the coefficients for one of the variables. Let us focus on the variable \(y\). The coefficients are 9 and 12. The least common multiple of 9 and 12 is 36.

To eliminate \(y\), we multiply the first equation by 4 and the second equation by 3:
\[4 \times (8x + 9y) = 4 \times 128 \implies 32x + 36y = 512\]
\[3 \times (5x + 12y) = 3 \times 131 \implies 15x + 36y = 393\]

  1. Solving for x

Now, subtract the second modified equation from the first:
\[(32x + 36y) - (15x + 36y) = 512 - 393\]
\[17x = 119\]
Dividing both sides by 17, we find the value of \(x\):
\[x = \frac{119}{17} = 7\]

  1. Solving for y

Now that we have \(x = 7\), we can substitute this value back into the first original equation to solve for \(y\):
\[8(7) + 9y = 128\]
\[56 + 9y = 128\]
Subtract 56 from both sides:
\[9y = 128 - 56 = 72\]
Dividing by 9:
\[y = \frac{72}{9} = 8\]

  1. Conclusion and Verification

To verify our findings, we substitute \(x = 7\) and \(y = 8\) into the second original equation:
\[5(7) + 12(8) = 35 + 96 = 131\]
The values satisfy both equations. Thus, the solution is \(x = 7\) and \(y = 8\), or \(\boxed{x=7, y=8}\).

Examples

This type of system of equations is very useful in everyday life, such as when buying items in bulk. For example, if you bought 8 pencils and 9 notebooks for a total of \$128, and a friend bought 5 pencils and 12 notebooks for a total of \$131, you can use these equations to find the unit price of each item. Solving systems of linear equations like this allows you to determine individual costs and plan your budget effectively when dealing with multiple combined purchases.

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