- Kadeem correctly multiplied the second equation by 5 to get \(5x + 5y = 125\), which is a valid step to align the coefficients for the elimination method.
- Kadeem's mistake occurred during the subtraction of the equations: he incorrectly calculated the new coefficient for \(y\) and the difference of the constants.
- The correct subtraction should yield \(9y = 119\), whereas Kadeem obtained \(-1y = 131\).
- The correctly solved system is \(y = \frac{119}{9}\) and \(x = 25 - \frac{119}{9} = \frac{106}{9}\).
- \(\boxed{9y = 119}\)
Explanation
- Analysis of Kadeem's Correct Step
The system of linear equations is given by:
1) \(5x - 4y = 6\)
2) \(x + y = 25\)
Kadeem's goal is to use the elimination method to solve this system. The first step he performed is to multiply the second equation by 5.
The second equation is \(x + y = 25\).
Multiplying both sides by 5: \(5(x + y) = 5(25)\), which results in \(5x + 5y = 125\).
Kadeem performed this step correctly. This is a valid algebraic manipulation, as multiplying both sides of an equation by the same non-zero constant preserves the equality.
- Identification of Kadeem's Error
Kadeem's goal was to eliminate \(x\) by subtracting the first equation from the second equation.
The transformed system is:
1) \(5x - 4y = 6\)
2) \(5x + 5y = 125\)
To eliminate \(x\), he should have subtracted the first equation from the second equation:
\((5x + 5y) - (5x - 4y) = 125 - 6\)
\(5x + 5y - 5x + 4y = 119\)
\(9y = 119\)
Kadeem, however, arrived at \(-1y = 131\). Comparing this to the correct subtraction, he likely incorrectly combined the coefficients of \(y\) (calculating \(5 - 4\) instead of \(5 - (-4)\)) or made an error in signs during the subtraction of the equations. Specifically, instead of correctly calculating \((5y) - (-4y) = 9y\), he appears to have simplified the subtraction incorrectly. Additionally, he calculated the difference of the constants as \(125 - (-6) = 131\) instead of \(125 - 6 = 119\), confirming an error in the subtraction process.
Examples
Elimination is a powerful tool in real-world resource allocation. For example, if a company produces two products, A and B, where the total cost is constrained and the relationship between the quantities produced of each is known, solving a system of equations helps managers find exactly how many units of each product to manufacture to stay within budget. Ensuring that every step of the algebraic manipulation, such as distributing constants and distributing negative signs correctly, is crucial for obtaining the right production levels and avoiding financial waste.