Verified Answer :

  • Multiply the equations to align the coefficients of \(y\): \(8x + 6y = 66\) and \(15x + 6y = 108\).
  • Subtract the first from the second to isolate \(x\): \(7x = 42\), giving \(x = 6\).
  • Substitute \(x = 6\) into the first equation to solve for \(y\): \(24 + 3y = 33\), yielding \(y = 3\).
  • The solution is \(x = 6, y = 3\), expressed as \(x = \boxed{6}\) and \(y = \boxed{3}\).

Explanation

  1. Analysis of the System of Equations

The problem asks us to solve a system of two linear equations in two variables:
\[4x + 3y = 33\]
\[5x + 2y = 36\]
To solve this system, we can use the method of elimination. This involves manipulating the equations so that one of the variables is eliminated when we add or subtract them.

  1. Elimination Strategy

We want to eliminate \(y\) from the equations. To do this, we find a common multiple for the coefficients of \(y\), which are 3 and 2. The least common multiple is 6.
Multiply the first equation by 2:
\[2(4x + 3y) = 2(33) \implies 8x + 6y = 66\]
Multiply the second equation by 3:
\[3(5x + 2y) = 3(36) \implies 15x + 6y = 108\]

  1. Solving for x

Now, subtract the first resulting equation from the second one to eliminate \(y\):
\[(15x + 6y) - (8x + 6y) = 108 - 66\]
\[7x = 42\]
Dividing by 7, we find the value of \(x\):
\[x = \frac{42}{7} = 6\]

  1. Solving for y and Verification

Now that we have \(x = 6\), we substitute this value back into the first original equation to find \(y\):
\[4(6) + 3y = 33\]
\[24 + 3y = 33\]
Subtract 24 from both sides:
\[3y = 33 - 24\]
\[3y = 9\]
Divide by 3:
\[y = 3\]
We can verify this in the second equation: \(5(6) + 2(3) = 30 + 6 = 36\), which is correct. Therefore, the solution is \(x = 6\) and \(y = 3\).

Examples

Systems of equations are used daily to optimize resources. For example, if you are buying supplies for a club and spend \(\$33\) for 4 notebooks and 3 pens, and later spend \(\$36\) for 5 notebooks and 2 pens at the same rates, you can set up the system \(\begin{cases} 4x + 3y = 33 \ 5x + 2y = 36 \end{cases}\) to determine that each notebook (\(x\)) costs \(\$6\) and each pen (\(y\)) costs \(\$3\). This helps in managing budgets and making informed purchasing decisions.

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