Solve the simultaneous equations.
$\begin{aligned}
7x + 2y &= 27 \\
4x + 3y &= 21
\end{aligned}$
$x = \quad y =$
We are given a system of two linear equations in two variables \(x\) and \(y\):
(1) \(7x + 2y = 27\)
(2) \(4x + 3y = 21\)
Our goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously. This represents the intersection point of two lines in the 2D plane. We will use the elimination method to solve this system.
To eliminate the variable \(y\), we can make its coefficients in both equations opposites. The coefficient of \(y\) in equation (1) is 2, and in equation (2) is 3. The least common multiple of 2 and 3 is 6.
We multiply equation (1) by 3:
\(3 \times (7x + 2y) = 3 \times 27\)
\(21x + 6y = 81\) (3)
Next, we multiply equation (2) by 2:
\(2 \times (4x + 3y) = 2 \times 21\)
\(8x + 6y = 42\) (4)
Now, subtract equation (4) from equation (3) to eliminate \(y\):
\((21x + 6y) - (8x + 6y) = 81 - 42\)
\(21x - 8x + 6y - 6y = 39\)
\(13x = 39\)
Dividing both sides by 13, we find:
\(x = \frac{39}{13} = 3\)
Now that we have \(x = 3\), substitute this value into one of the original equations to find \(y\). Using equation (1):
\(7(3) + 2y = 27\)
\(21 + 2y = 27\)
Subtract 21 from both sides:
\(2y = 27 - 21\)
\(2y = 6\)
Divide by 2:
\(y = \frac{6}{2} = 3\)
We have found \(x=3\) and \(y=3\). Let us verify these values in equation (2):
\(4(3) + 3(3) = 12 + 9 = 21\).
This matches the right-hand side of the second equation. Thus, the solution is correct. The final values are \(x=3\) and \(y=3\).
Systems of linear equations are used extensively in economics, engineering, and logistics to find an equilibrium point. For instance, if you are planning a production schedule for two different products, you can set up equations based on resource consumption (like raw materials or machine hours). Solving the system reveals the specific quantities of each product that perfectly utilize your available resources. In this case, finding \(x=3\) and \(y=3\) is like determining the optimal production quantities for two items given the constraints imposed by two different manufacturing processes.