For the system of equations x + y = 1 and 4x + 5y = 7, find x and y so that you can determine 4xy.
The correct answer and explanation is:
Let’s solve the system of equations: {x+y=14x+5y=7\begin{cases} x + y = 1 \\ 4x + 5y = 7 \end{cases}
Step 1: Express one variable in terms of the other
From the first equation: x+y=1 ⟹ y=1−xx + y = 1 \implies y = 1 – x
Step 2: Substitute y=1−xy = 1 – x into the second equation 4x+5(1−x)=74x + 5(1 – x) = 7
Simplify: 4x+5−5x=74x + 5 – 5x = 7 (4x−5x)+5=7(4x – 5x) + 5 = 7 −x+5=7- x + 5 = 7
Step 3: Solve for xx −x=7−5- x = 7 – 5 −x=2 ⟹ x=−2- x = 2 \implies x = -2
Step 4: Find yy using y=1−xy = 1 – x y=1−(−2)=1+2=3y = 1 – (-2) = 1 + 2 = 3
Step 5: Calculate 4xy4xy 4xy=4×(−2)×3=4×(−6)=−244xy = 4 \times (-2) \times 3 = 4 \times (-6) = -24
Explanation:
We started by isolating one variable, yy, in the first equation. This is a common technique for solving systems of linear equations. Substituting yy into the second equation allows us to solve for xx in terms of constants only.
After substitution, the second equation becomes a single-variable linear equation which is straightforward to solve. We find x=−2x = -2, then back-substitute to find y=3y = 3.
Finally, the problem asks to find 4xy4xy, not just xx and yy. Multiplying xx and yy and then by 4 yields −24-24.
This approach is simple and effective for any system of two linear equations. It ensures a clear step-by-step solution that avoids errors and helps understand the relations between variables.