Solving System of Equations and Comparing Quantities

The question provides a system of two linear equations with two variables, x and y. We are asked to compare the value of x (Quantity A) with the value of y (Quantity B).

The given equations are:

1. \( x + y = 10 \)
2. \( y = x - 2 \)

Quantity A is \( x \).

Quantity B is \( y \).

Solving the System of Linear Equations

We can solve this system using the substitution method because the second equation already gives us an expression for \( y \) in terms of \( x \).

Substitute the expression for \( y \) from equation (2) into equation (1):

\( x + (x - 2) = 10 \)

Now, we solve this equation for \( x \):

\( x + x - 2 = 10 \)

\( 2x - 2 = 10 \)

Add 2 to both sides of the equation:

\( 2x = 10 + 2 \)

\( 2x = 12 \)

Divide both sides by 2:

\( x = \frac{12}{2} \)

\( x = 6 \)

Now that we have the value of \( x \), we can substitute it back into either equation to find the value of \( y \). Using equation (2) is simpler:

\( y = x - 2 \)

Substitute \( x = 6 \):

\( y = 6 - 2 \)

\( y = 4 \)

Comparing Quantity A and Quantity B

We found the values for \( x \) and \( y \):

• Quantity A = \( x = 6 \)
• Quantity B = \( y = 4 \)

Now, we compare these two values:

\( 6 \) versus \( 4 \)

Since \( 6 \> 4 \), Quantity A is greater than Quantity B.

Let's summarize the values in a table:

Based on the comparison, Quantity A is greater.

Revision Table: System of Linear Equations

Additional Information: Solving and Comparing Variables

When faced with questions asking to compare quantities defined by algebraic equations, the key is often to solve for the specific values of the variables involved. Here are some points to remember:

• Always try to isolate the variables to find their numerical values.
• Systems of linear equations can be solved using various methods like substitution, elimination, or graphing. The choice of method depends on the structure of the equations.
• Once the values are found, perform the direct comparison as required by the question. Pay attention to whether the question asks if one quantity is greater, smaller, equal, or if it's impossible to determine.
• An "impossible to determine" outcome usually arises when the comparison depends on other unknown factors, or when the system of equations has infinite solutions or no solutions, making specific values for the variables impossible to find. In this problem, we found unique values for x and y, allowing for a direct comparison.